Modulated photoemission spectroscopy: Application to Au

SURFACE

SCIENCE 37 (1973) 355-385 0 North-Holland

MODULATED

PHOTOEMISSION APPLICATION

W. D. GROBMAN IBM T. J. Watson Research

Center,

Publishing Co.

SPECTROSCOPY: TO Au*

and D. E. EASTMAN Yorktown

Heights,

New York 10598,

U.S.A.

We describe modulated photoemission spectroscopy, in which an internal (sample) parameter such as temperature, or an external (apparatus) parameter such as wavelength is varied. A general formalism is developed for modulated photoemission spectroscopy and then illustrated using temperature modulated photoemission spectra and yields for Au obtained in the N 6 to 11.6 eV photon energy range. Modulated s-p band photoemission data are described in terms of photoemission critical points in order to explain the nature of the modulated structures in the s-p band region and relate experiment to energy band thresholds obtained from a recent band calculation for Au. Application of the formalism to modulated d-band emission leads to a method for extracting d-band deformation potentials. For example, we find that the upper d band edge moves upward with respect to EF at a rate of 2 to 6 x 1O-4 eV/K. Modulation of the quantum yield is described and our measurements are compared with modulated optical data.

1. Introduction Recent modulation techniques have significantly extended the usefulness of modulation spectroscopy as a tool for understanding the electronic structure of solidslv2). However, the systematic development of such techniques for photoemission spectroscopy, in particular physical or “internal” modulation, has received virtually no attentions). The purpose of the present paper is twofold: to develop some of the formalism describing the effects that contribute to modulated photoemission spectra and yields, and to then interpret temperature-modulated data for Au using this formalism. Photoemission spectroscopy of solids involves measurement of the kinetic energy distribution and total number (i.e., yield) of electrons emitted from an irradiated surface, these electrons having been excited from the occupied valence bands and core levels to high-lying conduction band states by monochromatic photons of energy ho. These energy distributions determine the one-electron energy levels of states contributing to tz2(CO)for excitations to final states lying above the work function @. Such information about the absolute locations in energy of the initial and final states has previously been * Supported in part by the U.S. Air Force Office of Scientific Research under Contract No. F44620-70-C-0089.

355

356

W. D. GROBMAN

AND

D. E. EASTMAN

shown to be of great value in understanding the nature of the valence and conduction band structure in many solids4,6). Analogous to the relation between unmodulated photoemission spectroscopy and optical studies, modulated photoemission spectroscopy can determine the absolute energy levels contributing to some of the features in modulated optical spectra5). In contrast to “external” modulation (where a property of the apparatus such as wavelength is varied), “internal” modulation, in which a physical property of the sample is modulated, can determine the deformation potentials’) of particular one electron energy states. As in the case of optical measurements, critical points in k space contribute structure in photoemission spectras-11). Modulation of photoemission energy distributions (PED’s) can enhance such features and thus help to understand the electronic structure of solidsrs,ls), e.g., via comparison of data with theoretical energy band calculations. Modulation of the quantum yield (the total number of photoemitted electrons per incident photon), is related to both modulation of the optical constants and to modulation of transitions into final states above the work function cutoffs). While we present a formalism for describing modulated photoemission spectroscopy and describe many of the important effects, our discussion is limited in scope, due in part to space considerations and the limited availability of experimental modulated photoemission spectra. Also, the formalism we develop considers only the case of modulation of a scalar parameter 5. Modulation of a vector or tensor property of the solid contains new information related to the symmetry of the states involved and is not considered in the present paper. Finally, experimental photoemission spectroscopy is constrained in the types of modulation methods that can be used by the requirements of uncontaminated sample surfaces (ultrahigh vacuum prepared) and the need to measure the energies of the emitted electrons. An outline of this paper is as follows. In section 2 we review the widely used “three-step” model of the photoemission process which we then use as a basis for describing scalar modulation effects in photoemission. This section also reviews the nature of structure in photoemission spectra at critical points in k space. The remainder of this paper applies some of the concepts developed in section 2 to the interpretation of temperature-modulated data for Au, which were obtained from the experiment3) described in section 3. The structure arising from modulation of the work function and Fermi distribution function is discussed in section 4. Au is a noble metal, with distinct s-p band (near the Fermi level Er) and d band regions contributing to the photoemission spectra. For the s-p band emission region, a description of modulated structure in terms of critical point concepts is

MODULATED

PHOTOEMISSION

SPECTROSCOPY

357

given in section 5. In section 6, discussion of modulation effects in the d band region proceeds from a different point of view, due to the large number of critical points contributing elements of structure. Section 7 develops and applies the formulas describing modulation of the photoelectric yield to data for Au. Finally, some of the major points described in the paper are summarized in section 8. 2. Modulated photoemission spectroscopy:

model, concepts and formalism

model49 1411s) of photoemission is used In this section the “three-step” to determine modulation effects on photoemission spectra and yields. We will mainly consider “internal” modulation, i.e., modulation of a physical parameter of the sample, although we also briefly consider modulation of an “external” parameter, one describing the experimental conditions (i.e., wavelength, electron energy, analyser acceptance energy). We also discuss photoemission critical points and the nature of their contribution to modulated spectra. 2.1. REVIEW OF PHOTOEMISSION MODEL The three-step model of photoemission assumes that the photoemission process can be described as the sequential occurrence of (1) optical excitation of electrons at energy Ei to states of energy Ei + hw, (2) transport to the surface (during which time elastic and inelastic loss processes occur), and (3) escape through the surface barrier into vacuum. We will use notation and results from this model given in ref. 4. We start with the general equation giving S(E,, ho), the number of photoemitted electrons per incident photon is) of energy Aw, these electrons having been excited from initial energy Ei measured with respect to EF the Fermi energy S (El, CO)= [ 1 - R (CD)]TeE (Ei + Aw)

f (Ei)[1 - f (Ei+ ho)]

Ef x

P,

(Ei,

w)

dE’ P, (E’, CD)K (Ei, E’, a)

+

s Ei

.

(2-l)

The terms in curly brackets describe the distribution of optically excited electrons per absorbed photon as well as the distribution of inelastically scattered ones. The Fermi distribution functions f (E) guarantee filled and empty initial and final states, respectively, while Teff contains the effects of electron transport to and escape from the surface. The factor containing the reflectivity R(w) is needed to normalize the emission to the number of

358

W. D. GROBMAN

AND

D. E. EASTMAN

incident photons. This normalization is not used in ref. 4, but is needed here as it enters the expression for the modulated yield. The particular form of T,, depends on several assumptions. A general form is

where the first term describes escape through the surface barrier and the second term describes transport to the surface. LX(O) is the optical absorptivity, I(E) the mean free path, and V, the inner potential for the solid. For the case of specular surface scattering4)

(2-2)

where Tf is the surface transmission probability for excited electrons at energy E above E,, assuming they are traveling normal to the surface. C, is a factor which corrects for the fact that the excited electrons have momenta which are randomly distributed with respect to the surface norma14). The first term within the curly brackets in eq. (2-l), P,(Ei, CO),gives the distribution of optically excited (primary) electrons and is normalized to unity : cc

s

dE, f (Ei) [I - f(Ei

+ Ao)] ‘, (Ei, 0) = 1

(2-3)

The second term inside the curly brackets gives the distribution of inelastically once-scattered “secondary electrons.” K(E,, E’, o) is essentially the distribution function for secondary electrons of energy Ei produced via electron-electron scattering and consequent electron-hole creating processes by an optically excited primary electron of energy E’> Ei. This term can be generalized, for example, by the inclusion of scattering due to surface and volume plasmons, which are important in simple metals. Also, in insulators this term would be modified by the addition of contributions from phonon scattering, which is unimportant for secondary electron production in metals4). The distribution function P,(Ei, a) for direct (k conserving) optical transitions is derived by starting with the dielectric loss function ~~(0)

MODULATED

given by

cs

e2

&2

PHOTOEMISSION

d3k12.P,,,J2

(Q) = __-

nm202

f(E,)

SPECTROSCOPY

[l - f(E,

359

+ h)]6(.&-En-h),

nn’

I~ap”“42 c2 (co) CYZ ds s IV,‘ (En, - &)I ’

(2-4)

where the second integral in eq. (2-4) is over the surface defined by the 6 function in the first integral. In eq. (2-4), e and m are the electronic charge and mass, 2 is the polarization vector of the incident light, E,(k) is the energy of the nth band in the solid at crystal momentum k (in the reduced zone scheme), and P,,,(k) is the momentum matrix element between bands n and n’ at k. P, (Ei, a) is obtained from eq. (2-4) by picking only those transitions which occur from initial energy E,,(k)=E,, and by normalizing to unity [seeeq. (2-3)]4).Thisisdonebyinsertingtheadditional6function6(E,(k)-Ei) into the first k space integral in eq. (2-4) and dividing the resulting expression by Ed. The resulting k space integral contains two 6 functions that confine the integration to the line of intersection of the surfaces defined by (2-5a)

E”(k)=Ei, E,,(k) Transforming

f(Ei) [l-

the volume

f

(Ei

+

integral

hw)l‘0

= e’J’(Ei)

- E,(k)

CEi,

to an integral

along this line, one obtains

w> W,“.12

[l - J’(Ei + AU)] 7rm202a2 (0)

where L denotes

(2-5b)

= ho.

cs nn’ L

d1 lV,E, x V,E,.] ’

(2-5b)

the line defined by eq. (2-5).

2.2. NATURE OF PED CRITICALPOINTS Modulated photoemission spectroscopy is analogous to modulated optical spectroscopy in two important respects: (1) modulation enhances structure arising from (or near) critical points17) in the Brillouin zone, and (2) if an internal parameter of the sample is modulated, “deformation potentials” for bands or band pairs can be obtained, where “deformation potential” denotes the change in a band energy per unit change in the modulation parameter. The two spectroscopies are different, however, due to the different dimensionalities of the k space region sampled in the two cases [compare eqs. (2-4) and (2-6)] and due to the absolute energy information contained in

W. D. GROBMAN AND D. E. EASTMAN

360

S(Ei, 0). As a consequence, qualitative differences occur in the nature of the critical points and information obtained using the two spectroscopies. We will summarize here, following Kaneii), the different types of critical points for ~(0) and S(E,, w). The nature of the information obtained by modulating S(Ei, w) and its relation to modulated Q(W) will then be discussed. We emphasize, however, that in practice much strong structure in photoemission spectra arises from regions in k space that are not mathematical critical points [the same behavior in s2 (0) has also been pointed out]. As a consequence, such regions can contribute many of the features seen in modulated PED’s, features which are not strictly included in a critical point analysis. As we will show later when we consider the case of Au, critical point concepts are most useful in analysing the emission from s-p bands. In the case of d bands, the large number of bands in a small energy range makes it difficult to associate particular critical points with individual structure elements in modulated spectra. The basic idea is that ~~(0) is related to the area of the k space surface defined by eq. (2-5 b), the one electron density of states p (Ei) at energy Ei is related to the area of the surface defined by eq. (2-5a), while S(E,, QJ) is related to the length of the line defined by the intersection of these surfaces. The equations defining critical points and their classification [following Kaneri)] are listed in table 1. p( Ei) and ~(0) possess three-dimensional critical points designated ECP and OCP whose analytic character is wellknown. For Q(O) it is of the form ~]o-c+,]~+b on one side of a critical value of w ( =wO) and it is constant on the other side. Fig. la sketches schematically the four types of three-dimensional critical points. In the case of s2, these occur at a point k, where V,E,, and V,E,, are parallel. Critical points in S(E,, 0) (as a function of Ei) occur for those Ei for which the surfaces defined by (2-5a) and (2-5b) are tangent at a point k,. If o is now varied, the k, and Ei at which critical behavior occurs changes, TABLE Nature Distribution function

PW e2 (w) S(Ei, 0)

I

of critical points for different Condition defining k space critical point

(1) V,En=O (2) V,(E,z-En)=0 (3) V,(Ens -En)

x V,E,=O

distribution

functions

Critical point designation and abreviation

Energy critical point (ECP) Optical point (OCP) Two dimensional critical point (CPa) (Occurs when either (1) or (2) hold or, in general, when the surfaces defined (2-5) are tangent)

by eqs.

MODULATED

a)

Fig. 1.

PHOTOEMISSION

b)S(Ei,w=canst.)

E2K4J)

361

SPECTROSCOPY

c )dS/d<

Schematic illustration of the nature of critical point structure in (a) &Z(O), (b) S(EI, w), and (c) aS(Ei,w = const.)/ae.

and k, follows a critical line [designatedll) CP,] in k space. If one plots those Ei corresponding to critical points on CP, as a function of o, one obtains lines which are designated by Kane as “&w images” of critical lines. The form of S(E,, o = const) for the three types of two-dimensional critical points is shown schematically in fig. lb. Two of these types produce structures in S(Ei, w=const) of the form SC

SL0,

Ei

>

E,

Ei < E,

or

Ei < E, Ei > E,

(2-7)

while the third type is of the form S = u + (SJZ)

In (Ei - E,I,

(2-8)

where S,, the strength of the singularity, is discussed in ref. 11. Modulation of an external parameter of the system, r, will modulate both E, and S,, leading to two contributions to W/St. These may contribute similar amounts to modulated structure once the d-function or (E-E,)-’ singularities in the derivatives of eqs. (2-7) and (2-8) are broadened by experimental and lifetime effects. Fig. lc shows schematically the shape of the broadened modulated versions of eqs. (2-7) and (2-8). If a structure such as those sketched in fig. lc is identified in a modulated spectrum, then the values of Ei and o at which it occurs give the absolute energies of the initial and final states contributing to the structure [via eqs. (2-5a) and (2-5b)]. Because many OCP’s and the associated critical frequencies w,, have already been identified for many materials via modulated Q(W) measurements, we consider now the possibility of using modulated photoemission spectra to determine the absolute energies of the initial and final bands at oO. Kane has shown that every OCP lies on a CP, line, so critical features in a PED will occur for some Ei for o near wO. The possibility of actually ob-

W.

362

D.FROBMAN

ANDD.E.EASTMAN

taining En and E,,, near a critical frequency o,,, however, depends on several factors. Some of the important cases are listed below : (I) Ei (crit)+tio, < #: Then the work function cutoff does not permit determination of En and Ens near the critical point at oO. (2) If Ei (crit) + ho, > @, oO, E, (crit) could correspond to an OCP but not an ECP. Then for o near wO, for the singularity of M, or M, type, (i.e., not a saddle point), and in the region where ~~(~)=a+b/cu--u~~(~, the PED has approximately constant amplitude and the higher and lower energy edges of the PED [Kane’s upper and lower critical points of type u and 1 (see fig. 1b)] are separated by a distance in Ei proportional to ]w -w,,lf. Thus in this case we would see in the modulated spectrum a set of features whose strengths were approximately unchanging approaching each other in initial energy E, at separation proportional to ]w---o,]+ as w-+ oO. (3) wg, Ei (crit) correspond to both an OCP and an ECP. This should only happen when required by symmetry. Then, if the singularities for both the ECP and OCP were of the M, or M, type (i.e., not saddle points), one possibility is a peak narrower than experimental resolution for a range of w near wO. Then a spectral feature would be obtained whose width is determined by the resolution of the apparatus and hole-lifetime broadening, and whose area diminishes as ]o--w~]~ as w approaches wg. In both cases (2) and (3) one can obtain both the initial and final energies E, and E,. contributing to Q(O) at w,, from modulated PED’s obtained for w near oO. Consideration of these effects in more quantitative detail or for different combinations of parameters is beyond the scope of the present paper. Finally, we note an important feature of PED critical points which is qualitatively different than for Ed critical points. The three types of critical points in S(E,, o= const) [see eqs. (2-7) and (2-8) and fig. 1b] are symmetric or antisymmetric functions about E,. Thus their derivatives, even when experimentally broadened, have a peak or zero crossing at E, independent of the broadening. This is in contrast to the case of e2 (o), which is not symmetric about a critical point oO. Near wO, the lifetime-broadened structure in modulated Ed has a peak position which is separated from u+, by an amount dependent on the broadening’s). 2.3.

GENERAL

MODULATION

FORMULAE

FOR

RESPONSEOF

S(E,,W)TOAN

INTERNAL

PARAMETER

An “internal” parameter r is now introduced; “internal” denoting the modulation of sample properties rather than properties of the apparatus as < is variedlg). In this section we will consider those modulated properties of the sample which result in modulation of the photoemission energy dis-

MODULATED

PHOTOEMISSION

SPECTROSCOPY

tribution (PED) or yield, Y(o). At the end of this section we also discuss briefly modulation of an “external” parameter. The variation of S(E,, w) [eq. (2-l)] upon varying an implicit modulation parameter 5 by an amount St is given by 20) _ 6pfJ PO +

dE’ PO (E’, O) K(Ei,

p.

(2-9)

E’, W)

s The first term describes the modulation of the number of photons absorbed while the second term describes the modulation of the work function (and thus escape probability) and the optical absorption depth. The third term (used for metals) describes the change in the energy distribution (Fermi statistics) near E, upon temperature modulation. The variation 6P, in the energy distribution of optically excited electrons is the term of most interest, for the position and amplitude of structure in &PO(Ei, CO) relates directly to the electronic energy band structure and to the magnitude of aE,(lc)/at (i.e., to the “deformation potential”). In the following paragraphs we discuss the individual terms in eq. (2.9) briefly. Following sections of this paper will then apply these results to the interpretation of temperature-modulated photoemission data for Au. We note that eq. (2-9) does not include the broadening of structure caused by a combination of lifetime broadening and experimental resolution. Such effects will be important in comparing experiment with the contribution from the various terms in eq. (2-9) in sections 4 through 7. 2.3.1. The effect of 6R The first term in eq. (2-9) is the modulated reflectivity. It depends on o but not Ei and so modulates the total emission intensity [the quantum yield Y(o)], but does not contribute structure in the modulated energy distribution 6S( Ei, CO).In section 7 we will use measurements of 6R/R by others5) in conjunction with eq. (2-l) in analyzing our measurements of 6YjY for Au. 2.3.2.

Modulation of T,,

The second term in eq. (2-9), although a function of Ei, is a smooth and slowly varying one4). Modulation of T,,(E) [g’iven in eq. (2-2)] occurs principally through modulation of CC(O), which is related to &sZ(w), and which contributes to the modulated yield, and through modulation of the work function @. The latter variation can in principle contribute structure in 6S, for near threshold Tef is of the form T,,(E,

+ hw) =

” const. x (ho + Ei - @),

E,<@-ho, Ei > @ - hw,

(2-10)

364

W. D. GROBMAN

so that variation form

AND

D. E. EASTMAN

of @ can result in a sharp edge in S&‘/S near threshold

STe, =

” - const.

E,<@-h&3, x (a@/@) St, Ei > Q?- ho.

of the

(2-11)

2.3.3. Modulation of the Fermi distribution function The term Sf(Ei) in eq. (2.9) contributes new informationer) only in the case of temperature modulation, in which case in describes changes in emission for Ei near EF due to changing occupation of the initial state E,,(k). No term in Sf(Ei + tzw) appears since the final state is always above @, which is many kgT above EF (k, is Boltzmann’s constant and T is temperature). It is important to note that there is no term in Sfproportional to SE, since EF is the energy from which all quantities experimentally accessable in photoemission spectroscopy are measuredar). As i is varied, emission from EF occurs at a fixed value of all parameters characteristic of the electron energy energy analyzer for constant fiws1). Thus Sf in eq. (2.9) only contributes to SS when temperature is the internal parameter modulated, and then it contributes in the form ST df Sf =aTST= x (2-12) 2(1 +coshx) -T ’ x = (E, - E,)/k,T. Sf is an odd function about x=0 which contributes positively to SSfor Ei > EF and negatively for Ei O. That is, it describes extra emission from states above EF as T is increased and a corresponding decrease in emission below E,. 2.3.4. The contribution

of

PO(Ei, co)

The most interesting effects in SS(&, o) occur through the term SP, in eq. (2-9), which is directly related to the modulation of the energy band structure and momentum matrix elements. For emission from s-p bands, this term can give the absolute energy positions of both the occupied and unoccupied state along photoemission critical lines as discussed in detail in section 2.2. In the case of d band emission, interpretation of the data in terms of critical points is in practice difficult, for the large number of occupied bands in a narrow energy range gives rise to a large number of critical point features close together in the Ei-o plane. In this case structure in SP, can be interpreted primarily in terms of the modulation of the position of the bands with respect to E,, and can in principle be used to determine the coefficient of the change in position of prominent d band features with respect to variation of the modulation parameter 5.

MODULATED

In calculating

PHOTOEMISSION

SPECTROSCOPY

6P, we rewrite eq. (2-6) in the form

(IT~~co~.s~ (w)/e2) PO (Eiy w)=

11 dl

(P,,“, =

C In”,,

(2-l 3)

nn’

nn’

L

where li*P,,.12 cp ““’ = (v&“(k)

x V&n+)l’

(2-14)

and the integration in the line integral Z”“, proceeds along the k space line L defined by eqs. (2-5). The variation in one of the Z”“, occurring as a consequence of the variation of 5 by St consists of two main parts: &ZII”’= 81”’ nn’ 9 ““’ + &Z(2)

(2-15)

where sZ$) arises from the variation of I”“, in eq. (2-13) with the path of integration L unchanged, while 61 ‘,‘,,!arises from the change in I”“, occurring upon integrating the unvaried 4,,, along the new path L+ 6L. (l) describes those effects arising from the change in ampliPhysically, SZ”“, tude of emission, while 61”“. (2) describes the effect of changing thresholds and should thus contribute most of the sharp structure in 6P,. We obtain 2 2,

2Re[(e*Pnn,) (i-6Pnn,)1 IG x ‘&I

L -

,;‘f”;‘2,3 (G, x G,s)-(G,,

x 6G, - G, x SC”.)},

(2-16)

n’

and

dl V. (G,, 6E, - G, &),

=

(2-17)

s L

where 64, = %

St,

These general formulas can be used to understand

G, = V,E, (k),

6G” =

(2-18)

will prove to be useful in two areas: First, they general features of the modulation such as the

W. D. GROBMAN

366

AND

D. E. EASTMAN

relation between modulation of internal and external parameters (see section 6). Second, they provide formulae for calculating 6P, using existing computer codesss) which at present already contain integrations along the line L. 2.3.5.

Modulation

of Ei or o

In the case where the external parameter Ei is modulated, U’,‘,!=O [this occurs for modulation of any external parameter”*)] and eq. (2- 17) become+)

Sri;! = - [ di V.(Gn, - G,) 6E,.

(2-19)

J L

In the case of wavelength the form 26)

modulation

(2) -_ 61,,*

SZ’,‘,!zO also24), but now M’,‘,! is of

dl V-6’,< &II.

(2-20)

I

L

Comparing eqs. (2-19) and (2-20), we speculate that a comparison of i$and w-modulation may in simple cases be used to sample the initial and final states separately. Because 61 “) Rnidescribes the effects of changing ~~~e~~o~~~ for transitions, while &I\:! describes changing amplitudes for structure existing already in P,, Ei, or w modulation will be [and has been in the past12)] quite useful in comparing calculated energy bands with experiment (see section 6). The new information given by internal modulation is the mugnitude of I??&/&‘<{the “deformation potential” for internaf parameter 5). This derivative can depend sensitively on the assumptions made in energy band calculations. Its experimental value can then be of use in testing the physical assumptions of a theory. Another kind of information, obtained using nonscalar27) internal modulation (e.g., strain modulation), is information about the symmetry of certain optical transitions. At present this information has been obtained only from modulated optical experiments to any significant extent, but the successes there should encourage the future use of these techniques in photoemission spectroscopy. 3. Temperature moduIated spectra and yields for gold : experimental technique The remainder of this paper will apply some of the concepts and results of section 2 to the interpretation of temperature-modulated data for gold3). In this section we give a description of the experimental method”) used to obtain temperature-modulated photoemission energy distributions (PED’s) and yields. The data will be presented in figs. 4,6, and 7 in sections 5,6, and 7. Temperature-modufated PED’s and quantum yields were obtained using a

MODULATED

PHOTOEMISSION

SPECTROSCOPY

367

standard photoemission spectrometer consisting of a spherical retardinggrid electron energy analyzer in an ultrahigh vacuum system with a LiF window and a l-m normal incidence monochromator with a hot filament Hinteregger-type H, lamp. PED’s and their energy derivatives were obtained using a standard ac bridge technique with a lock-in amplifier synchronously detecting the ac component of the photocurrent4). Electrons were photoemitted from a -2 cm2 area on Au films deposited in ultrahigh vacuum on the front of a 1 urn thick gold foil about 6 cm2 in area which was supported at the edges by a mica frame. The back of the foil was coated with colloidal carbon (aquadag) and heated by light focused from a 100 W Zr arc lamp. Temperature-modulated PED’s were obtained at discrete photoelectron kinetic energies by observing the fluctuation of the PED signal at a fixed retarding voltage as the heating light was turned on and off at a rate of about + cycle/mins*). A similar procedure was used to obtain the temperature-modulated yield2s). The on/off temperature swing of the foil was estimated to be about 50 + 25 K from measurements of its resistence. Temperature-modulated reflectance measurements usually use resistive heatingITs) of the sample to modulate T, either using an ac current through the sample or through a separate heater in the substrate. In a photoemission experiment, magnetic fields associated with such a heater could modulate the photoelectron trajectories (unless unusual care was taken to fabricate the heater in a special geometry) and cause an unwanted signal synchronous with the thermal modulation. For this reason infrared heating was used as described above. To ensure that the heating light did not produce any photocurrent, the heating light source was focused onto the back of the sample substrate with a Pyrex lens which acts as a low pass filter that transmits only photons whose energy lies below the photothreshold for the graphite backing of the sample substrate. Use of a gold backing foil prevented problems due to thermally-stimulated diffusion of substrate atoms into the film being studied. (Problems of this sort arose when a Cu foil was used as the substrate.) Calculations were performed to determine the temperature modulation rate and amplitude that could be obtained with a heating light such as the one described above. A simple optical system with a Zr arc lamp source can produce a heating power input Pi in the -0.1 to N 1.0 W range into the (high absorptivity) lampblack coating on the back of the 1 urn thick Au foil. If the Au foil is in poor thermal contact with the remainder of the apparatus, so that radiation cooling dominates conductive cooling, then the temperative swing AT is determined only by Pi and can be well over + 100°C a total swing ~200°C about four times that achieved here. Such an amplitude will occur as long as the heating-cooling period is long enough to permit radiation heating (or cooling) to supply the heat determined by

368

W. D. GROBMAN

ANV

1). E. EASTMAN

AT and the heat capacity of the foil, and calculations indicated that for our sample this would occur with a heating (or cooling) time constant of N 1 set (N 1 Hz modulation frequency). In practice we only obtained ATx 50°C and heating/cooling time constants of N 10 to 20 set were typical. This degradation of AT and heating time constant from the ideal was caused by a significant thermal contact to the mica support frame. Photoemission spectra and their temperature derivatives were obtained for hv=7.7, 8.6, 10.2, 11.1, and 11.6 eV and are shown in fig. 4 in section 5. The energy resolution of the spectrometer is z-O.25 eV as determined from the sharpness of the cutoff in S(E,, o) at Er. The signal-to-noise ratio in this experiment is such that the sensitivity is typically of the order %$‘SN 10p3. In our experiment it waslimited primarily by uv lamp drift, due to the tow frequency (-$cycle/min) of modulation. Better thermal isolation of the sample and use of a second lock-in2s) amplifier to detect the temperature-modulated signal could improve this sensitivity. This sensitivity can also be expressed in terms of the magnitude of modulation-induced energy band shifts that can be detected. A few x 10W4 eV/K is a typical temperature coefficient for the d band to E, separation (see section 6) and typical structures in the d band region of the spectra in fig. 4 are of the order of I part in IO* of the unmodulated energy distribution and correspond to a modulation temperature swing of 50 K or energy shifts of the order of lo-* eV, so energy band shifts of the order of a millivolt could yield detectable signals. Derivatives of the spectra with respect to Ei were also obtained by measuring the second derivative of the photocurrent with respect to retarding voltage using second harmonic detection in the lock-in amplifier. TWO of these E,-modulated spectra are presented and discussed in section 6. The temperature-modulated yield from 6.0 to 11.6 eV was also obtained, by detecting the temperature-induced fluctuation in the output of an electrometer (which measured the total photocurrent) in a manner similar to that used for obtaining &S/C?&? Fig. 7 presents our resuhs for A Y(o)/Y in section 7, where they are compared with reported measurements5~ of temperaturemodulated reflectivity for Au. 4. Fermi distribution and work function effects Thermal modulation causes extra emission for E, > EF as temperature is increased [see eq. (2-12)], leading to a positive peak just above EF. Convolution of eq. (2-12) with a gaussian distribution of full width 2r=O.25 eV (approximately equal to our experimental resolution) results in a function whose width is determined by our experimentat resolution (which is greater

MODULATED

PKOTOEMISSION

SPECTROSCOPY

369

than the thermally-induced spread), while its peak area gives information about the temperature modulation amplitude. The total emission near EF, obtained by differentiation ofeq. (2-2), is the sum of two terms:

a3 =S2g~+f(Ei)aT’

as aT

(4-l)

E~=EF

where S zz s/f(Ei). When c?,!?/aTis negative, it decreases the positive extra emission for Ei >Er given by eq. (2- 12) and increases the height of the negative peak for Ei < EF. The opposite effect occurs for aS/dT> 0. We corrected the observed positive peak for Ei > 8, for this effect in the modulated spectra for hv= 7.7, 8.6, 10.2, and 11. I eV in fig. 4 in section 5. This resulted in values of AT consistent with our estimate ATz50+25 K obtained from measurements of the resistance charge of the Au foil. Another important modulation effect involves the sharp cutoff in the final states for optical transitions caused by [l -f(Ei + Iiw)] in the expression for E~(CIJ) [eq. 2-4)]. Modulation of the bands by an internal parameter (or w modulation) can cause large effects in 6~~ when EF is near the final state for transitions from lower states. Many of the features in de,/aT for Au have been attributed to such transitions. This Fermi distribution effect on &sZ can contribute features in 6Y (section 6) but not in gS(E,, CD). The temperature-induced work function change AQix (a~/aT)AT should add a constant to the modulated spectra 63’ near threshold (for Ei 2 @ - fro). Such a constant would not appear with the same amplitude in E,-modulated spectra, and a comparison of aSjaT with dS/BE, near threshold (see fig. 6 in section 6) would show a constant difference in this energy range. However, such an effect is not seen in fig. 6 for the following reasons. Typical values for a@/aT are experimentallyzg) -0.2 x 10e4 eV/K for (211) Ta, or z -0.05 x 10m4 eV/K for a theoretical calculationsa) for a simple metal (jellium model). Temperature coefficients for the energy bands in Au are typically of the order of 2 x 10e4 eV/K (see section 6). Since for Aw = 1I. 1 and 11.6 eV the slope of Teff near threshold is ZC+ of the slope of S(Ei,U) at the upper d band edge at Ei = - 2.1 eV, (see fig. 4, Aw= 11.1 and Il.6 eV in section 5), one would estimate from the value of d@/aT for Ta a signal due to 6T,, of - l/20 that of the peak in 6S in fig. 4 at E, = - 2.1 eV. This signal magnitude is close to the detectability limit in our experiment. The lack of a discernable contribution to SS from temperature modulation of the work function in this experiment is then easily understood: it results from the smallness of the temperature coefficient of the work function for typical materials. From the numbers given in this section it appears that the effect of a@/dT on 6s will be unimportant for many metals.

370

W.D.GROBMAN

ANDD.E.EASTMAN

5. Modulated emission from s-p bands The critical point description of S(Ei, w) (section 2.2) is useful as a starting point for a description of modulation effects in materials (and energy ranges) where the critical points are well separated. For Au in the range -2.0 eV< 5 Ei ~0, emission occurs only from a single band whose energy dispersion E,(k) is qualitatively similar to the lowest “s-p” band in simple metals. Emission from this “s-p” band into two higher bands then contains information about a few critical lines (for the range of hw of interest here) which occur at values of E, which are well separated at a given value of o. Emission from the d bands of Au (Eis -2.0 eV) contains structure due to a large number of criticallines which contribute to S(Ei, w) in such a complex way that the formal theory of critical points is not easily applied in practice. In this case, which is treated in section 6, the discussion of modulation effects proceeds without reference to critical point concepts. In section 5.1, we discuss from both the experimental and theoretical points of view the nature of simple critical points in S(E,, 0) for Au (in the s-p band region). Section 5.2 then discusses the modulated spectra in the s-p band region (-2.0 eV< Ei GO). 5.1. CRITICAL POINT FEATURES IN THE s-p

BAND

EMISSION

FROM

Au

We start with the experimental evidence for simple critical point structure in Au for -2.0LEi,<0. In fig. 2 we have plotted S(E,, co) for Au for 10.2
MODULATED

PHOTOEMISSION

SPECTROSCOPY

11.2 1 ; i 102

I--I-

I

i; -6 -5 -4 INITIAL ENERGY (eV)

Fig. 2. Photoemission energy distribution spectra for Au plotted as a function of initial energy EI measured with respect to EF( = 0). These data were taken from ref. 31.

conduction band) as described below. The plot of Ei versus w for this edge, which is given in fig. 3 (crosses) lies on the line L2 (lower energy edge of transitions into band 8). In order to relate these transitions to specific electron energy states, we refer to a recent relativistic APW calculation of the bands of Au by Christensen and Seraphinsa). They give the band structure for Au, plotted along symmetry lines as well as along non-symmetry directions, sampling most of the Brillouin zone. If we consider vertical transitions from states at EF, the lowest photon energy FU at which they can occur into band 7 (for any of the bands plotted in ref. 32) is 3.0 eV (El), along the symmetry line L-W, while the largest value of hw at which they can occur (into band 7) is 8.6 eV (E2), near k=(& 2, 0)27r/a. Transitions into the next highest unoccupied band (band 8) from E, do not start until fio=9.5 eV (Es), along the symmetry line T-K. Thus, as fro is increased, ref. 32 predicts that there is a range of fiw in which one finds no transitions into bands 7 or 8. Consideration of the

W.D.GROBMAN

372

33

-2 Ei -INITIAL

AND

D.E.EASTMAN

-I

O=EF El

ENERGY BELOW E, (eV1

Fig. 3. Features in the E&m plane related to critical structure in the s-p band region as described in section 5. Crosses are the position of critical point features in S(EJ, w) obtained from ref. 31. Circles and small squares are similar features obtained from refs. 33 and 13 respectively. Ex, EZ and Ea are thresholds for transitions from band 5 at EF to bands 6 and 7, obtained from ref. 32. In regions I and II strong emission occurs from band 5 into bands 6 and 7 respectively. In region III, d band emission dominates. The region below the line marked “Cp = 5.1 eV” is experimentally accessable only for cesiated samples.

bands in ref. 32 for other values of initial energy Ei < Er gives the same picture: at a given value of E, < EF, emission into band 7 turns on (as ttw is increased) at some low threshold (in the 3 to 4 eV range, which is experimentally accessable only for cesiated Au) and turns off at a higher value of tiw. Then no emission occurs from states of that initial energy E, for a range of values of increasing tto until the threshold for new transitions into band 8 is reached. This behavior is qualitatively seen in the data. The experimentally determined E,-o positions of the edges in S(E,, o) for Au from ref. 31 (fig. 2) are

MODULATED

PHOTOEMISSION

SPECTROSCOPY

373

plotted in fig. 3 (crosses), as well as the position of such edges from refs. 33 and 13 (circles and squares respectively). (The latter data were obtained using cesiated Au with a lower work function.) These Ei-w edge positions fall on the three lines shown, separating the Ei, w plane into different regions. In regions I and II (shaded) the s-p band emission is significantly stronger than in the unshaded regionss4). Thus at constant Ei (defining a vertical line in fig. 3), for increasing hw the intensity of emission varies qualitativelyss) as deduced from the band structure diagrams in ref. 32. Quantitatively, the lines Ll, Ul, and L2 (obtained from the experimental edge positions), which we are interpreting as Ei-w images of Kane’s critical lines, can be extrapolated to E,=O, which they intersect at thresholds of Ao=3.4, 8.4, and 9.8 eV respectively. The ref. 32 threshold fro values for these points were noted above to be about 3.0,8.6, and 9.5 eV. These theoretical thresholds at Ei=O (EF)are denoted in fig. 3 by arrows labeled Ei,E,, and E,. In order to strengthen the arguments presented here, we have performed a model band calculation in which relativistic 4-OPW energy bands were calculatedss) which approximated the Au s-p bands in the region of interest. Photoemission spectra for these bands, calculated using the Gilat-Raubenheimer method 23), gave “square-box”-like emission features (emission with sharp edges of height comparable to the average emission) whose widths decreased to zero at the thresholds in Ao obtained from the band structure. 5.2. MODULATED

SPECTRA FROM THE s-p BANDS

In fig. 4 we give S(E,,w) (dashed) and 6S(Ei, w)= (&S/aT)AT (solid lines) for Aus), obtained as discussed in section 3, for ATx 50 K. Here we plot the modulated and unmodulated emission intensities as a function of initial energy Ei,measured relative to EF (= 0)at fixed fiw. In the s-p band region, - 2.0< Ei< 0 we identify strong negative peaks in 6s centered at Ei= -0.1,-0.8,- 1.2 and - 1.4 eV for Fro= 8.6, 10.2, 11.1, and 11.6 eV respectively, and positive peaks at Ei= -0.43and -0.7 eV for Aw= 11.1 and 11.6 eV. The Ei-o positions of these negative and positive peaks in 6s (Ei, co)are plotted in fig. 3 (squares surrounding “ - ” and “ + ” signs). The negative peaks in 6s lie on the line Ul while the positive peaks lie on the line L2, and are obviously the modulation enhancement of the emission edges occurring along Ul and L2. The sign of 6s along these lines can be understood as follows. In ref. 32, the effect of AT>0 on the s-p bands is primarily that of the increasing lattice parameter (decreasing Brillouin zone volume) which moves s-p bands closer together (see section 6 for further discussion of this point). Decreasing band separation lowers the Aw positions ofthe lines L2 and U 1 in fig. 3, causing increased emission at L2 and decreased

374

W.~.~R~BMAN

0;

AND

1

U.E.EASTMAN

--

hb’=8.6&

c

;

j

hU=10.2eV

hZ.‘=li.l

I’

eV

1

‘IA

”

z

/--‘,

1 hU=llCeV

/(

\ I

_A

“5

_& INlTtAL

Fig. 4.

Photoemission

_j

-2

-i

O=E,

/

ENERGY ieV)

spectra (dashed) and temperature-modulated given as a function of initial energy Ei.

spectra (solid) are

MODULATED

PHOTOEMI~IO~

SPECTROSCOPY

375

emission at U I, as observed. Another statement of the effect can be obtained by differentiating the form of S(E,, 0) at a critical point of type I as given in eq. (2.7) and fig. lb: S (I$, co) = $6 (Ei - E,, (co)),

w3

i%

-=iw

SL

(?T

a(Ej-E,)+

=L aT

@@i--o),

(S-1) f5-2)

where the function 8 is 0 for negative argument and 1 for positive argument. The threshold value of El decreases as T is increased, for the reasons given above, so dE,/aT in eq. (5-2) is negative and a positive peak at E, appears. The discussion of an upper critical point is similar. The second term in eq. (S-2) describes the change in amplitude S, with temperature. The first and second terms in eq. (5-2) correspond to ZILt? and #,,L! in eq. (2-15), which correspond to changing thresholds and amplitudes respectively. In fig. 4 we always see negative emission in &S/ST between the critical point peaks, coming from a negative value 0f as,/aTin eq. (5-2). It is instructive to compare (internal) temperature-modulated emission in the s-p band region with (external) o- and &modulated emission. The emission intensity of the “square boxes” decreases as w is increased due to the factor [u&s2 (w)]-’ in eq. (2-6). For w modulation, this factor will cause negative w-derivatives in the s-p band region between critical point peaks, just as in the case of thermal modulation. For E, modulation this factor does not enter. Further, gr’,‘,! is zero for this case as noted before. Thus only the peaks at thresholds will occur, but aS/i3Ei will be near zero otherwise. This effect is seen in the region - 2.0< Ei GO eV in the curves of aS/aEi given in fig. 6 in section 6. 6. Modulation of d band emission spectra Emission from the d bands of a transition or noble metal such as Au has been shown experimentally *,6,12) to depend on the joint energy distribution P, [eq.(2-6)], so that one might expect temperature modulation of S(E,, W) in the range of Ei corresponding to d band initial states to depend on modulation of both the initial (d band) state E,(k) and final (s-p band) state En.(k). However, as previously reporteda), the high density of states in the d band region causes modulation of the initial state to dominate tip,, (Ei, co). In this section we use the formalism developed in section 2 to clarify the manner in which initial state modulation dominates SS in the d band region, and present data illustrating these effects. These results will be used to indi-

376

W. 0. GROBMAN

AND

D. E. EASTMAN

cate a method for rather simply obtaining d band deformation potentials from internally modulated PED’s. We present a model which contains many of the features of temperature modulation of the d bands, and will show what effects in SP,(& w) it produces. The starting point is the KKR picture of the dependence of the d bands on crystal potential. The result of this analysiss’) is that the d bands shift and their overall width changes as the lattice dilates (for example as a consequence of increasing temperature) without changing their shape. A more precise statement is that a d band energy can be written in the form E,(k) = E, (a) + a%,&(k),

(6-E)

where a is the lattice constant, E. is the center of the d bands, To is an a-independent width parameter, and F,,(k) is a function of k which is independent of a. Upon differentiating eq. (6-l) with respect to a and using eq. (6-l) to eliminate To we find

(6-2) Eq. (6-2) has been tested by using eigenvalues for the d bands of gold in ref. 32. There E,(k) was computed for Au using the relativistic APW method, for both a “normal” volume lattice and a dilated lattice (A4azO.013). Using tables 2 and 5 in ref. 32 we have plotted [E”(k) (dilated lattice) -E,(k)] versus E,,(k) for 38 d band energy eigenvalues along symmetry directions (see fig. 5). The points lie close to a straight line of slope -0.065. This indicates that eq. (6-2), which predicts that SE,(k) is a linear function of E,(k) (for arbitrary k) with this slope for the lattice dilation used in ref. 32, is an excellent first approximation to describing d band shifts with lattice constant. Ref. 32 indicates that the lattice dilation effect is the major effect of temperature on the d bands, so that the modulation parameter 5:= a for the present analysis. The contribution to SP, from bands n and n’ due to changes in the photoemission line integral, SI\t?, is of the form [see eqs. (2-17) and (2-l8)f

where G,, and G,, are the gradients of E, and E,* [see eq. (Z-18)]. Eq. (6-2) does not give all the information necessary to specify SE, (the change in initial state d band energy) in eq. (6-31, for eq. (6-3) assumes energies are referenced to E, (as this is the nature of the photoemission measuring process), while eq. (6-2) is referenced to the muffin-tin zero. For this reason SE, is obtained in terms of dE,/aa given in eq. (6-Z) by subtracting from it dE,lda (also referenced to the muffin-tin zero)

MODULATED

PHOTOEMlSSlON

377

SPECTROSCOPY

0

SLOPE = -0.065 g 6 B & 5

-15-

5 -2oA % I? ’

-~ -25 \o

-30

-100

Fig. 5.

8 Oo 0 R \ 0 1 I 300 400 RYDBERGS) \

A---L 0

__i_ 100 200 E,, (I$ (MILLI

i

500

d Band energy shifts A&(k) upon lattice dilation by 1.3%, calculated in ref. 32, are plotted versus En(k) for 38 d-band eigenvalues given in ref. 32.

(b-4) The d state shifts SE,, and SE,,, (the change in the final state, s-p band energy) are approximately equalse). However, jG.,l/lG,l is typically 2 5 to 10 since G,, is a d band energy gradient while G,, is an upper s-p band gradient. Therefore, a good approximation to ZiI(n2n! in eq. (6-3) is (6-5) We now compare this result with the corresponding results occurring upon modulating the electron energy analyser acceptance energy Ei. We found in section 2 that for this case UC,‘,! [eq. (2-16)] was zero and SI’,z,! was the same as eq. (6-3), but with 6E,, = &En. = -SE,. Again IG,,[ is of the order of 5 to 10 times larger than IG,l so that for Ei modulation dl V.G,,f 6E,.

sr$J Z s

(6-6)

Comparing eqs. (6-5) and (6-6), we find thatfor any small range of Ei (for which SE,, is approximately constant) al.../& is proportional to -dZn,,/dEi

378

W.D.GROBMAN

AND

D.E.EASTMAN

for d band emission. This proportionality implies that sharp structure in &S/D will correspond in shape with sharp structure in -aS/dEi in any narrow range of Ei, since SI(,k!, which appears in the T- but not in the E,-derivative should only contribute a more slowly varying background (see section 2). These comments, along with eqs. (6-5) and (6-6), also show that quantitative comparison of sharp structures in aS/aT with those in -aS/aE, at a given value of Ei can give the deformation potential of the d band states at that value of E, (measured with respect to EF). We will use experimental data to illustrate this technique by applying it to the states at the top of the d bands in order to experimentally determine their deformation potential. The experimental modulated spectra dS/dT and -dS/dEi are shown in fig. 6 for the two highest photon energies studied, 11.1 and 11.6 eV. Use of ___~ -

(a)+f

GOLD

-1

(b)-$$. ,

hv= Il.1 eV

I

1 q 0

0

I -(a)

I

L

-6

I

-5

_..LIl-~-_ii -4 INITIAL

Fig. 6.

X$/W and -

-3

-2

ENERGY

-I

O=EF

(eV)

a,S/aE are given as a function 11.6 eV.

of inital El for ho == 11.1 and

MODULATED

PHOTOEMISSION

SPECTROSCOPY

379

these photon energies brings the widest energy range of d band structure above the photothreshold. For both values of ttw we see the same elements of sharp structure at any particular initial energy Ei 2 2 eV in the temperature modulated spectra as in the E,-modulated spectra, as expected from the discussion above. (For a given photon energy, the positions of corresponding structure elements lie within about 0.2 eV of the same initial energy.) Also, the temperature modulated spectra contain no additional sharp structure beyond that appearing in the voltage-modulated curves, indicating that the contribution of SI’,‘,! to the temperature-modulated curves is indeed relatively smooth (or small), as suggested above. Comparing the most prominent feature in the two modulated spectra (at either value of Ao), the sharp positive peak at Ei x - 2.1 eV, we find a deformation potential for the upper d-band edge of + 2 to 6 x 10m4 eV/K (with respect to EF) for AT= 50-L-25 K. (The range of values given here corresponds to the uncertainty in AT). This deformation potential can also be estimatedss) from ref. 32 as about 1.O x 10e4 eV/K. While internal modulation is required to determine deformation potentials, if all that is required is enhancement of structure for comparison of experiment with theoretical band positions, then our results show that the more easily performed (external) Ei modulation leads to the same position and shape for sharp structures that is given by the more difficult internal modulation techniques. 7. The modulated yield In the case of internal modulation, it is convenient to work with the yield per incident photon Y(o) [rather than the more usual4) yield per absorbed photon], i.e. the total number of emitted electrons per incident photon, since the reflectivity is modulated. Y(o) is given by cc

Y(0)

=

s

dE, S (Ei, w).

(7-l)

-02 Using eq. (2-l), this becomes

Yi =

c,

s

dEi T,(E) al(E) f (Ei) [l - f(E)1 EF

s El

dE’ P,(E’,

O) K (E;, E’, W) .

(7-2)

380

W.V.GROBMAN

ANV

D.E.EASTMAN

In eq. (7-2) we used eq. (2-2) for T;, and replaced 1(E) in the denominator by f becausea) al $1 for Au. Also, we introduced the unmodulated value of a denoted 8, and defined Es Ei +hw. Because ~14 1, Yi is just the yield per absorbed photon at constant (unmodulated) a(o). Upon differentiating eq. (2-2) with respect to an exteral parameter (5) we obtain AY

AR

Y

R

Acr 1 + .cc i_+--~+

AZ’, Yi -*

(7-3)

Modulation of the yield occurs via two basic processes: (1) modulation of the optical constants changing either the total fraction of photons absorbed [first term in eq. (7-3)] or the fraction absorbed within the photoelectron escape depth (second term); (2) modulation of AYi. The second effect can be described as follows. Yi is the fraction of the contribution to eZ(w) [see eqs. (2-4) and (2-6)] coming from optical transitions to final states of energy E,,s(k)-E, -6tiw greater than the work function @, weighted by T,(E) i(E). The first two terms in eq. (7-3) contain information obtainable from optical measurements, while AYJY, contains information concerning transitions with final states above threshold. A Yi contains contributions from three sources: (I) changing Q,could cause the contribution of transitions to states near @ to change, (2) final states could move further above threshold, or (3) the amplitude of P, at some E, above threshold could change. The last two effects are discussed in sections 2 where they are described by the integrals ZJLz! and &lb:! [eqs. (2-17) and (2-16)]. Whereas (1) can be important for some semiconductors and insulators, it should be negligible for metals (see section 4). Contribution (2) is weighted most heavily in the region where T,(E) 1(E) is varying most rapidlyg) (i.e., for @=~E~-i-fiw~@+ ,- 3 eV), while (3) is weighted most heavily where ~~(~~~(E~ is largest (.&+-ho>@+ -3 eV), although in both cases significant contributions occur for all values of Ei -I-hw > @. To obtain an idea of the values of fro at which AY contains a large contribution from AYi, AY/Y measured in a photoemission experiment should be compared with the first two terms on the right hand side of eq. (7-3) detemined from optical data as follows. If AR/R is measured for the material of interest, then Ak, the modulated component of the imaginary part k of the complex index of refraction (n+ik) is found from AR/R by KramersKronig analysis in the usual way 1s). Then Aala is found in terms of Ak from c(/k,

Au c

SI

::

Ak k’

where c is the speed of light. In the case of Au for which values for A&*(W) have already been pu-

MODULATED

PHOTOEMISSSON

381

SPECTROSCOPY

blisheds), we have used an alternate procedure. We have found Acr/ccin terms of Ak and k by using the published curves for AR/R and AQ and by eliminating An from the pair of equations38) AE2(m)=2kAn+2nAk, AR

(7-4)

4(n2 -k2-l)An+8nkAk (7-5)

R=[(n+1)2+k2][(n-l)2+k21.

We obtained n and k from experimental values of tzl and ~~ in ref. 39. We compare AY/Y with [-AR/R+ (Acc/~) (1 -elf)‘1 in fig. 7 (where we used i= 30 A). In order to compare the temperature-modulated optical data obtained by Scouler (for which AT- 1 or 2 K) with our measured value of A Y(w)/ Y(w) (f or which AT-50 K), we had to correct for the ratio of the temperature modulation amplitudes in the two experiments, both of which are uncertain by N 50%. A ratio of 50: 1 [corresponding to AT(present paper) z 50 K and AT (Scouler) x 1K] gave the curve in fig. 7 which agrees reasonably well with A Y/ Y in regions where they possess similar structures. The largest difference between the two curves occurs between tiwz6 and -7.5 eV, which is then the region in which AYi should be largest [see (7-3)]. At higher values of Aw, AYi/Yi appears to be small. We speculate on

L

I

6

I

7

I

9

I

IO

II

Lc",

Fig. 7. Our results for A Y/ Y (squares) and 50 x [ - (AR/R) + Acu/a(l + CU’)]obtained from ref. 5 are plotted as a function of photon energy ho.

382

W. D.GROBMAN

AND

D. E. EASTMAN

two possible contributions to AYi in the 6-7.5 eV region. Near fiw= 7 eV, fig. 5 of ref. 32 shows transitions occurring from the top of the d bands (band 5) to band 7 along the lines X-K and X-W near X. Along both of these lines, the band 5 to band 7 separation is approximately constant (parallel bands), so ~(0) may receive a significant contribution from this transition near Ao~7 eV. (These bands are not parallel over a wide range of k in the third direction, X-T, so the actual size of the contribution can only be obtained from a proper calculation.) The final states in this transition lie only a few tenths of an eV above @ (~5.1 eV) according to ref. 32. Increasing temperature (lattice dilation) will cause these final state bands to move downward in energy (see sections 5 and 6) where T, is smaller, which could contribute a negative value of AYi near hwx7 eV. Alternatively, (or simultaneously), AYi can be receiving contributions from s-p band transitions in the 6-7 eV range. Further measurements of 6S(Ei, w) for 6
8. Summary The present

paper

develops

a formal

(1) The determination of d band deformation potentials for transition and noble metal d band states as a function of their energy, and (2) the calculation of predicted modulated spectra using existing computational methods. The

MODULATED

PHOTOEMISSION

SPECTROSCOPY

383

temperature deformation potential for the motion of the upper d-band edge with respect to E, was found to be 2 to 6 x 10e4 eV, in comparison with about 1 x 10e4 eV as estimated from ref. 32. Fermi distribution effects were observed and were quantitatively related to the measured temperature modulation amplitude. Work function modulation effects on the measured spectra were not seen, and this observation was related to experimental and theoretical determinations of the size of the temperature coefficient of the work function. Finally, the relation between modulated optical data and the modulated yield was derived and used in comparing our measurements of dY/dT with previous temperature-modulated measurementss) of AR/R. Modulated photoemission spectroscopy experiments in which parameters other than T or Ei are varied, as well as the extension of these methods to simple metals, semiconductors and insulators should provide important new information for understanding the electronic structure of many solids. Acknowledgments We wish to thank J. Janak for the derivation of eq. (2-17), and for many helpful conversations. Also we acknowledge useful discussions with E. 0. Kane, S. Kirkpatrick, N. D. Lang, R. A. Toupin and A. R. Williams. References 1) Semiconductors and Semimetals, Vol. 9, Modulation Techniques, Eds. R. K. Willardson and A. C. Beer (Academic Press, New York, 1972). 2) M. Cardona, Modulation Spectroscopy, Suppl. 11 of Solid State Physics, Eds. F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1969). 3) W. D. Grobman and D. E. Eastman, Phys. Rev. Letters 28 (1972) 1038. 4) D. E. Eastman, Photoemission Spectroscopy of Metals, in: Metals, Vol. 6, Part 1, Eds. R. F. Bunshah (Wiley, New York, 1972) ch. 6. 5) Temperature modulated optical spectra for Au are given in: W. J. Scouler, Phys. Rev. Letters 18 (1967) 445. 6) N.V. Smith, CRC Critical Rev. Solid State Sci. 2 (1971) 45. 7) In this paper, “deformation potential” is defined as the change in a band energy per unit change of a modulation parameter. 8) EZcritical points are discussed in refs. 1,2,9 and 10. 9) L. Van Hove, Phys. Rev. 89 (1953) 1189. 10) D. Brust, Phys. Rev. 134A1337 (1964); J. C. Phillips, J. Phys. Chem. Solids 12 (1960) 208. 11) Photoemission critical points are analysed in E. 0. Kane, Phys. Rev. 175 (1968) 1039. 12) Extensive use of &modulation for enhancing structure to compare with band calculations has been made by N. V. Smith. See ref. 13 and references therein. 13) N. V. Smith, Phys. Rev. B 5 (1972) 1192. 14) C. N. Berglund and W. E. Spicer, Phys. Rev. 136 (1964) A 1030, A1044. 15) E. 0. Kane, in: Proc. Fifth Intern. Semiconductor Co&, Suppl to J. Phys. Sot. Japan (1966).

384

W.D.GROBMAN

AND D.&EASTMAN

16) We normalize emission to the number of incident photons (rather than the more usual number of absorbed photons) since the optical absorptivity can be modulated. 17) SeerefsStoll. 18) B. Batz, Thermal and Wavelength Modulation Spectroscopy, chapter 4 of ref. 1. 19) Mathematically, we now assume that En and En, in eqs. (2-5a) and (2-5b) are explicitly dependent on &, and k becomes implicitly dependent on e due to the constraint imposed by eqs. (2-5). For example, differentiation of eq. (2-5a) then gives + aE,/@ + + (ak/ay)*vE,(k) = 0. 20) We have dropped the term describing modulation of the secondary electron distribution in eq. (2-9) since it should not contribute structure in SS for metals. 21) If Q is modulated, then a term proportional to 8y(Er)/aEr (a peak centered at EF appears. 22) The derivation of eq. (2-17) is due to J. Janak, private communication. 23) Such an alogorithm for computing S(Ei, w) is described in J. Janak et al., Direct Transition Analysis of Photoemission from Palladium, in: Efectrunic Density ofStates, Ed. L. H. Bennett (Nat. Bureau of Standards, Special Publication 323, U.S. Govt. Printing Office, Washington, D. C., 1971) p. 181. 24) When 5 is an external parameter, E,&,En, in eqs. (2-S) do not depend explicitly on r. so SE, = 6Rzr = 0. 25) If, for example the external parameter <= Et, then differentiation of (2-5a) gives - a&/al + V&(k). The minus sign here compared with the + sign in ref. 19 is the origin of the - sign in eq. (2-19). 26) Eq. (2-19) contains G,&and Gnr since Ei appears in both eqs. (2-5a) and (2-5b). For w-mod, only 6En, # 0 so only G, enters eq. (2-20). 27) The case of non-scaler modulation is not considered by us here, but the formula in section 3.2 could be easily generalized to this case in many instances. 28) A second lock-in amplifier can be used to detect the temperature-modulated signal if its frequency is somewhat slower than that used to modulate the photocurrent (to pass through the RC filter in the first lock-in). 29) H. Shelton, Phys. Rev. 107 (1957) 1553. 30) N. D. Lang, Solid State Commun. 7 (1969) 1947. 31) D. E. Eastman and W. D. Grobman, Phys. Rev. Letters 28 (1972) 1327. 32) N. E. Christensen and B. 0. Seraphin, Phys. Rev. B4 (1971) 3321. 33) P. 0. Nisson et al., Solid State Commun. 7 (1969) 1705. 34) Emission is seen ex~rimentally even in the unshaded region in fig. 3 (between Regions I and II). The origin of such emission has never been identified. It could be due to surface photoemission“) or impurity-assisted non-k-conserving transitions. 35) Some of the characteristics of the “box-like” features in s-p band emission described here have been discussed previously. See S. Methfessel, Z. Physik 147 (1957) 442; R. Y. Koyamaand N. V. Smith, Phys. Rev. B2 (1970) 3049; and ref. 4, section II-D. 36) This calculation used the relativistic 4-OPW scheme described in J. R. Anderson and A. V. Gold, Phys. Rev. 139 (1965) A 1459. 37) J. M. Ziman, Proc. Phys. Sot. (London) 86 (1965) 337: V. Heine, Phys. Rev. 153 (1967) 673; J. Hubbard, Proc. Phys. Sot. (London) 92 (1967) 921; A. W. Williams, private communication. 38) This is a very approximate estimate, obtained by us by sampling the motion of Only a few s-p band eigenvalues near EF in ref. 32. 39) D. Beaglehole, Proc. Phys. Sot. (London) 85 (1965) 1007.

Discussion Question (by D. BEAGLEHOLE): I notice that your 10.2 eV curve looks different from the others in that here the modula-

MODULATED PHOTOEMISSIONSPECTROSCOPY tion data do not look like the derivative of the others. Do you have an explanation Speaker’s

of the unmodulated

spectrum

385 as they do in most

for this?

reply (by W. D. GROBMAN):

I do have an explanation,

and it brings out an important

point. The measured

spectra

are the product of two functions of initial energy Ei.One is the distribution of photoexcited electrons in the solid, whereas the other is the function, going to zero at the vacuum level (as Ei decreases), that describes transport to the surface and escape. Only the derivative of the first function is taken when the bands move, so the measured modulation spectra do not look simply like the derivative of the unmodulated one in the case of internal (physical) modulation. In the case of the EI modulation, however (an external modulation technique), the modulated spectrum is the derivative of the unmodulated one. Question (by G. E. JURAS): Stress-modulated photoemission spectra can be more decisive than thermomodulation measurements in identifying specific features of the band structule. What are the difficulties involved

in doing such measurements?

Do you plan to do any in the near future?

Speaker’s reply (by W. D. GROBMAN): Because photoemission gives the absolute energies of the states involved,

even thermally

modulated data can identify specific band structure features. Nonscalar (e.g. stress) modulation does give new information about the symmetries of the states involved. Modulation is more difficult in photoemission because you must restrict yourself to modulation methods that don’t modify the energy distribution of the emitted electrons. Stress modulation could be done, but I don’t have any present plans for doing that experiment.