Inverse photoemission from solids: Theoretical aspects and applications

Surface Science Reports 8 (1987) 1-41 North-Holland, Amsterdam

INVERSE PHOTOEMISSION FROM SOLIDS: THEORETICAL ASPECTS AND APPLICATIONS G. BORSTEL

and G. THORNER

Fachbereich Physik, Universitiit Osnabri~ck, D-4500 Osnabrfick, Fed. Rep. of Germany Manuscript received in final form 18 May 1987

The current status of inverse photoemission from clean crystals is reviewed with special emphasis on theoretical aspects and applications. The article focuses on general photoemission theory, the effective barrier potential at metal surfaces and the calculation of inverse photoemission spectra for both bulk and surface electronic states. In particular the information originating from inverse photoemission on Shockley and image-potential surface states, effective masses of mage-potential states and spin-split surface states at ferromagnets is discussed.

0 1 6 7 - 5 7 2 9 / 8 7 / $ 1 4 . 3 5 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )

G. Borstel, G. ThSrner /lnverse photoemission from solids

Contents 1. Introduction 2. Theory of inverse photoemission 2.1. General theory 2.2. Models 2.3, Surface contribution in the one-step model 3. Inverse photoemission from bulk states 4. Inverse photoemission from surface states 4.1. Surface states in multiple scattering theory 4.2. Examples: C u ( l l l ) and Cu(100) 4.3. Inverse photoemission and effective surface barrier in Cu(100) 4.4. Effective masses of image-potential surface states 4.5. Spin-split surface states at ferromagnets 5. Summary and conclusions Acknowledgements References

3 3 3 7 10 13 16 16 18 26 33 36 39 39 40

G. Borstel, G. Thi~rner/ InversephotoemissWnfrom solids

1. Introduction Inverse photoemission has become an important experimental tool to study the electronic structure of clean and adsorbate covered solids in recent years [1-7,87]. Inverse photoemission is a technique complementary to the now widely used angle-resolved ultraviolet photoelectron spectroscopy, since it may probe electronic states in an energy range not accessible to ordinary photoemission. Inverse photoemission involves the physical process where electrons with energy E~ impinge onto a solid and are de-excited by emitting photons with energy h~0. If Ef denotes the energy of the final electronic state, conservation of energy results in E i = E f + ho~.

(1.1)

The electrons impinging o n t o a solid occupy previously unoccupied unbound electronic states of the solid, i.e. E i lies above the vacuum level Ev~c of the sample. Since prior to de-excitation the final state must be unoccupied, E/ must lie above the Fermi energy E F of the solid. The region between the Fermi and the vacuum level is not accessible to ordinary photoemission and it is just in this energy range where inverse photoemission has its natural application. The s-p-like bulk bands of noble and transition metals, Shockley surface states, broken bond states at semiconductors, the minority spin d-states in 3d ferromagnets and antibonding orbitals of adsorbate molecules are interesting examples for electronic states between E F and Eva c. In order to interpret experimental inverse photoemission data it is necessary to have a reliable theoretical model to calculate the emission spectra. The description of various theoretical aspects of inverse photoemission and the application of the theory to bulk and surface electronic states in solids is the subject of the present contribution. The field of inverse photoemission is expanding very rapidly. In what follows we will concentrate mostly on one major application, namely inverse photoemission from clean metals.

2. Theory of inverse photoemission 2.1. General theory

In this section we present the general theory of inverse photoemission and specify the approximations which must be made to transform the problem into a form which allows for practical calculations. We follow closely the derivation given in ref. [88]. The transition probability per unit time w between two

G. Borstel,G. Th5rner/ Inversephotoemissionfrom solids (N + 1)-electron states of the same Hamiltonian H u+l denoted by I q ~ +1) and I g'vN+I) is given by Fermi's golden rule, if the perturbation A is small 8( E ; + , - e ; + ,

w=

-

(2.1)

E~ +1 and E Fu + a a r e the energies of the (N + 1)-electron states I g'tN+I ) and I q'FU+l), and hw denotes the energy of the outgoing photon. Defining annihilation and creation operators a S and a f for an electron in a one-particle state I q~f) with orbital energy e/, the coupling to the electromagnetic field A may be written in second quantized form

A = E ai, a'~a,

(2.2)

f,l with one-electron matrix elements a v = (,/,i Ia I q,z). Assuming that the initial many-electron state [x/,ff+l) may be represented as an antisymmetrized 9roduct of a one-electron excited state [@i) and the N-electron ground state • g ) , i.e.

%~+~)= a,+l%~).

E p +~ = ~ , +

Eg,

(2.3)

we obtain . laiY', a/ta * t+ asl~f~"+ 1) = 8 ( Eft+ ' _ EFU+, W= h2~ ( '~t'~

h~).

(2.4)

f,l

If ei is sufficiently large and I ~N+X) not too far from the ( N + 1)-electron ground state, the component of I q~i) in I Xr'u+I) F is almost exactly zero, a~l q'Fu+a) = 0, and a nonvanishing w requires i = l:

2

W= ~ - Ef A~ilvl-tg af~J; +1)

8(gp+l-g;+a-hog).

(2.5)

This may be written in the form

_ T2~r :~, X--,a.l~ul a I~'+1\/ :i\ o i :l ,~

w -

×

8(e, '<+' - e~' +' -

(qlFN+' af, lg'~:)ai,,,

h~0) (2.6)

and by summing over all possible initial and final states i, F we obtain for the total photon current Ip(hW) the final expression Ip(haa)-

E aTi Al/'(ei'.hl°) a/,,, i,f,/'

(2.7)

G. Borstel, G. ThOrner / Inverse photoemission from solids

where =

z (

-

F

+

× ( g'ff+ 1lay+ [q'g)

(2.8)

denotes the spectral function of the electron state. We will now give different approximations of eq. (2.7) in order to clarify the significance of the spectral function Aif,(e i - boa). The one-particle states I~f) may be chosen in such a way that the off-diagonal elements of Aff, become minimized. Assuming that these elements may then be neglected, we obtain

lp(hoa) - ~_, IAfi I 2 A//(e i - hco),

(2.9)

f,i

i.e. the observed photon current is essentially the spectral function Afi weighted by the one-electron transition probability I Afi} 2. Introducing the complex self-energy 2~(ey) of the electron state one has 1

A//(e,-hoa)= r

IIm E(ef) I

[ei_e/_h~o_ReN(e/)]2+[imX(e/)] 2.

(2.10)

The photon current Ip(hoa) will therefore exhibit Lorentzian shaped quasiparticle peaks at energies h~0 = e i - e / - Re E(ei) if IIm E(e/) I << I EI+ Re N(ef) I and Im N(ef) is a slowly varying function of energy. In the non-interacting limit E(e/) = 0 the spectral function A/y becomes 8(e i - e l he0) and consequently

Ip(h~o)- E laf, I 2 8(~,- ~:- h~0),

(2.11)

f,i

i.e. in this limiting case the spectrum exhibits discrete peaks with zero width at energies which exactly correspond to the one-electron energy differences Ei ~

Ef.

Since a discussion of 2f for different systems is outside the scope of this article, we will give only some general remarks on self-energy corrections to the one-electron result (2.11). For a simple metal with delocalized electrons the self-energy may be calculated in the jellium approximation. The real part of ,Y then turns out to be essentially a constant/*xc(n) right out to wavevectors k of approximately twice the Fermi wavevector kr, i.e. over an energy range of four times the Fermi energy E F. The constant/,xc(n) is just the exchange and correlation contribution to the chemical potential # of jellium with electron density n and is typically of the order of 10-20 eV for metallic densities. Since this constant part of Re 2; gives rise to a rigid shift in all one-electron energy levels, it can be omitted in (2.10). Of interest are then the self-energy correction Re N(k)-/~x¢(n) and Im ,~(k). Corresponding graphs are shown

G. Borstel, G. ThOrner / Inverse photoemission from solids

10

5- Re T(k}-P.×c

l

O cr kl_l

LU

I Ira Z (k)l

0

1 2 3 MOMENTUM (k/k F}

Fig. 1. Self-energy correction and imaginary part of the self-energy for jellium as a function of reduced quasi-particle m o m e n t u m [8].

in fig. 1 for a density n appropriate to A1. Both Re ~ ( k ) - / & c ( n ) and Im ~ ( k ) vanish exactly at the Fermi momentum, but otherwise are nonzero due to the possibility of electron-hole and plasmon excitations. These results show that even for simple metals one must expect nonzero self-energy corrections to the one-electron energy differences ei - e/ in (2.11), as one moves away from the Fermi energy. For metals with more localized d- or f-levels corrections due to ,Y tend to increase. Re ~ may then not be a slowly varying function of energy, as in fig. 1, but show a pronounced energy-dependence, which in turn may give rise to the occurrence of additional peaks (satellites) in Ip. The unoccupied bands near E F in frequently studied 3d and 4d metals like Cu, Ni, Pd and Ag are, however, mainly of s - p character and consequently the jellium results in fig. 1 are expected to hold for e > E v in these systems, too. In fact the self-energy correction in such materials has been found experimentally [9] to vary from E F to E F + 70 eV in a way qualitatively very similar to that shown in fig. 1. Similar remarks hold for insulators where, as a general rule, one finds Re < 0 for the occupied and Re ~ > 0 for the unoccupied electronic quasi-particle states. Fig. 2 shows as a typical example the self-energy correction Re ~(e) - ~(~t) for the insulator MgO calculated within the Penn model [10]. The self-energy behaves discontinuously across the gap and results in an energy correction of 0.8 eV at the bottom of the conduction band.

G. Borstel, G. ThSrner/ lnversephotoemissionfrom solids

0.1-

~LDA-Gap7~'J~

,21..

/

t.4

//

//

//

/

I

MgO

t,.,4 G.I or"

-01U -0.4 -0.2 Fig. 2. Self-energy

I

1

I

0 (E-it) (Ry)

I

0.2

I

0.4

correction in a typical i n s u l a t o r ( M g O )

[10].

2.2. Models It has been shown by Pendry [11,12] that there exists a close relationship between the electron current I e in photoemission from solids and the photon current Ip in the corresponding inverse photoemission experiment. In fact in analogy to (2.1) we have for the photoemission case 2~r W = T < ~/I;[A I~/*g) 2 ~}( E ; -- E g - - J~o) ), (2.12) where E ~ and Eft are the energies of the N-electron states I'/'0N ) and I g'~v > and h~a denotes the energy of the energy of the incoming photon. With the approximations ['tlZFN)= a f I~N-I>, E N = ~f'+- g N - l , (2.13) where ef denotes the energy of the photoelectron state and I q'~ -1> the Sth excited state of the ( N - 1)-electron system with energy E ~ -1, we obtain in analogy to (2.7) I~(h~o)- E aT,' A,,,(el-h~o) A/,, (2.14) f,i,i' where

A . , ( e z - boa) = E ( ~'gla~ [ ~ N - 1 ) ~ ( Ef-- hto -[- E 2 -I - f g ) S X < ~ 2 - 1 [a i [ ~ g >

now denotes the spectral function of the hole state.

(2.15)

G. Borstel, G. Thrrner / Inoerse photoemission from solids

In the independent-particle limit, where the spectral functions become &functions, the difference in I e and Ip is entirely due to the corresponding normalization factors, which have been omitted in (2.7) and (2.14). The differential cross sections generally contain a factor pe/vi, where pf is the density of states of the outgoing particles and u i the velocity of the incoming ones. Since these quantities are different for photons and electrons, Ip and I e differ by a factor of order a 2 [5,11], where a = e2/hc is the fine-structure constant. Thus the inverse photoemission cross section is about 5 x 10-s times the photoemission cross section. Since Iv and Ie are so closely related to each other, they may both be calculated within the same model. Neglecting off-diagonal contributions i ~ i' in the spectral function A , , ( e / - h~) one has for I~ the expression

Ie(ho~)- ~_.,(ep/IA~.,lepi)Ai,(~f-ho~ ) f

(
(2.16)

i

By introducing the Green's function G1 of the hole state via

ElePi) 3 ( e f - h w - e i )

(q~i[ = - l I m [ G ; ( e / - h ° ~ ) ]

(2.17)

i

we obtain in the non-interacting limit

I~( h~) - Im[ ~ ( q~AG~ A' [~,l ] .

(2.18)

For an energy-and angle-resolved experiment the state of the photoelectron at the detector may be written as I k,, el), where kll denotes the wavevector component parallel to the surface. Introducing the Green's function G 2 for the photoelectron we have l epf ) = G~- bkll, e/) and therefore in the non-interacting limit

I~( k,, e/ ) -

Im[(k,,,

q.IG] AG~ AtG~ Ikt,, ef) ] .

(2.19)

For the interacting system the common procedure is to retain (2.19) and to take into account only a renormalization of the propagators due to self-energy corrections in the internal interactions [13,14]. Since vertex corrections are then not taken into account, eq. (2.19) will result in the elastic part of the photocurrent only, calculated for an unscreened optical field. The coupling to that unscreened field is described by e

e2

A = ~--~mc( A .p + p . A ) - eCb + 2mc2

. A,

(2.20)

where A and • are the corresponding vector and scalar potentials. The term A • A in (2.20) does not contribute to one-photon processes and is neglected.

G. Borstel, G. ThOrner / Inverse photoemission from solids

Since V .A = 0 , ~ = 0 in the Coulomb gauge and A = A 0 c o s ( q - r ) - - A o in the visible and ultraviolet spectral region, where the optical wavelength 2~r/q is large compared to typical atomic distances, one has from (2.20) A = --~-eA 0 .p.

(2.21)

me

In actual calculations the omitted vertex corrections are sometimes included as an afterthought by calculating the screened optical field inside the solid from classical macroscopic dielectric theory and by taking into account inelastic losses via an empirical contribution to the imaginary part of the self-energy ~. The problem how to incorporate vertex corrections in an a-priori theory of the photoemission process in solids has been attacked recently [15,16], but so far these investigations seem not to have resulted in a practical calculational scheme. A practical calculational scheme on the basis of eq. (2.19), however, has been worked out for closely packed semi-infinite solids by Pendry and coworkers [14,17]. In this so-called one-step model of (inverse) photoemission the spectra are calculated in a formalism of independent quasiparticles with finite lifetimes in a semi-infinite crystal. In principle the Green's functions entering (2.19) must be calculated from the Dyson equation

["

~m + Vc(r)= -8(r-

1

J

E G(r, r', E) + d3r '' Y~(r, r", E) G(r", r', E)

r')

(2.22)

for the semi-infinite solid. Herein Vc(r ) is the Coulomb potential due to all charges in the system and ~(r, r', E) denotes the non-Hermitian, nonlocal and energy-dependent self-energy operator. In actual calculations within the one-step model one uses, however, a local approximation for the self-energy and regards X as a correction to the (real) exchange-correlation potential Vxc(r ) of the density functional theory [18]: ~ ( r , E) = Vxc(r ) +

BYe(r, E).

(2.23)

With this approximation the total effective potential entering (2.22) becomes complex via 8~ and has the form

V(r, E ) = Vc(r ) + Vxc(r ) + rYe(r, E).

(2.24)

To facilitate the calculation of the transition matrix elements, the total effective potential of the semi-infinite crystal is formally split into a bulk (MT) and a surface barrier (B) part: Re V(r, E ) = Re vMT(r, E ) + Re

V~(r, E).

(2.25)

Inside the solid Re V MT has the form of a bulk muffin-tin potential shifted in such a way that the spatially constant energy-dependent part V0(E) of

G. Borstel, G. ThOrner / Inverse photoemission from solids

10

Re V MT between the muffin-tin spheres becomes zero. Outside the topmost atomic layer Re V MT is supposed then to vanish. Im V MT takes a spatially constant energy-dependent negative value Voi(E ) inside the solid and is smoothly joined to Im V B near the topmost atomic layer. The muffin-tin approximation works very well for closely packed solids and for such materials one therefore expects inside the semi-infinite specimen Re V = Re V MT, Re V B = 0 . Outside the topmost atomic layer, on the other hand, one has Re V = Re V B, Re V MT = 0. Since both V MT and V B are complex via 8X in (2.24), there is no clear-cut distinction between electronic bulk and surface states in (2.22), i.e. this one-step photoemission model has the advantage of treating both bulk and surface states on an equal footing. The significance of V in the photon or electron current becomes obvious from the observation that in the electric dipole approximation (2.21) one has Afi = ~

e

ihe

(q~f I A° "P 1¢~ = ( ~ f - ~)mc (~f I A°" V V I ~,),

(2.26)

if I ffi) and [~y) are eigenfunctions belonging to the same single-particle Hamiltonian. The last expression shows that there exists no contribution to the current from those regions where the effective potential is constant. The current may then be split into a bulk part I MT originating from the rapid variation of Re V MT in the muffin-tin spheres and a surface barrier part I B due to the spatial variation of V B. The bulk part IeMv has been calculated by Pendry [14] and needs not to be repeated here. The surface part 17, however, requires a separate discussion. In Pendry's original formulation Re V B was simply taken as a one-dimensional step potential jumping from the bulk muffing-tin zero Vo(E ) to the vacuum level. In view of the fact that the observation of unoccupied surface states by inverse photoemission has become an area of great interest in recent years, this simple approximation appears now too crude. We will thus give in the next section some remarks on the surface contribution in the one-step model. It should be noted that (2.26) does not hold exactly for the energy-dependent effective potential V(r, E) in (2.24), since then in general 1 ~ ) and I'~f) do not belong to the same single-particle Hamiltonian [19]. This inconsistency stems from the neglect of vertex corrections in the basic formula (2.19) and a formalism to overcome this deficiency has recently been proposed [20].

2.3. Surface contribution in the one-step model For a clean and unreconstructed crystal surface the effective barrier potential VB(r, E) will take the form [21,22]

V"(r, E) = E V~a(z, E) e ig'*'', g

(2.27)

G. Borstel, G. Thrrner / Inoerse photoemission from solids

11

where g denotes a two-dimensional reciprocal lattice vector and vii is parallel to the surface. Self-consistent electronic structure calculations for solid metal films [23] have demonstrated that the (real) ground state potential VB(r, E = Ev) has a distinctly three-dimensional character. The periodic variation of V B for directions parallel to the surface is commonly referred to as corrugation. At the present level of accuracy in low-energy electron diffraction (LEED) and (inverse) photoemission experiments corrugation effects seem, however, to be of relative insignificance [24]. Thus all terms with g 4:0 are usually disregarded in (2.27) and V B is approximated in one-dimensional form,

V"(r, E)= V"(z, E).

(2.28) We now assume that the crystal occupies the halfspace z > 0 and the origin z = 0 is just at the center of an atom in the topmost atomic layer. For closely packed solids the muffin-tin part in (2.25) should hold right up to this topmost layer and we thus strictly demand Re VB(z, E ) = 0 for z > 0. Far outside the solid (z ~ - ~ ) VB(z, E) becomes real and approaches the vacuum level Evac- The zero of the energy scale in (2.25) and in all actual calculations within the one-step model coincides with the bulk muffin-tin zero V0(E), but for the sake of clarity V0(E ) is explicitly written out in this section. We use in the following atomic units (e = h = m = 1). VB(z, E) should then fulfil the following requirements: (i) near the topmost atomic layer it must go smoothly to VoMT(E) = V0(E ) + iV0i(E); (ii) far outside the crystal VB(z, E) must exhibit the correct asymptotic behaviour in z; (iii) V~(z, E) should be continuous and monotonous, dVB(z, E)/dz continuous as a function of z. The requirement (iii) is plausible since calculations of the (real) ground state barrier potential Vn(r, E = EF) in the framework of density functional theory give a continuous and monotonous behaviour in z. Density functional theory provides also a hint to the correct asymptotic form of Vn(z, E) for z--* - ~ . It may be shown rigorously [25] that far outside a metal surface VB(z, E = Ev) must vary as Evac + 0.25(z - zim) -1 + Va(z - zim) -2. Herein the (negative) z-2 contribution is due to exchange, whereas the (negative) z-1 image-potential term is only due to correlation [25]. It then seems plausible to require the same asymptotic behaviour also for Re VB(z, E) [26]. If one further assumes that the energy-dependence of Re VB(z, E) is entirely due to the energy-dependence of V0(E ) and connects V0(E ) to the asymptotic regime by means of a polynomial with minimal degree in z, one finds for Re VB(z, E) the form [27] 'Evac "~ 1( Z -- Zim

-1

"~ V I ( Z -- Zim)

-2

(Z < Z2 < Zirn),

ReV"(z,

E)=

%+°l(z-z2)+Vz(Z-Zz)2+°3(z-z2)3

v0(E) (z>z3).

(2.29)

12

G. Borstel, G. ThUrner / Inoerse photoemission from solids

Since the coefficients v; of the third-order polynomial are fully determined by the requirement (iii), Re V B exhibits four parameters, namely the position of the image plane zim, beginning and end (z 2, z3) of the polynomial region, and V1 in the exchange term. The precise asymptotic behaviour of Im VB(z, E) for z---) - o o is not known. It is obvious that IIm VB(z, E)] must approach zero at the vacuum side very rapidly [26]. To simplify the total barrier potential V ~ it is convenient to assume an asymptotic (z - Zim)-2 behaviour for Im Va(z, E). With the same additional assumptions that led to (2.29), Im VB(z, E) then takes the form [27] V2(z--Zim) -2

<

Im V"(z, E) =

< Z m),

w° + wl(z - z4) + w2(z - z4)2 + w3(z - z4)3

(2.30)

V0i(E)

This form of Im VB(z, E) exhibits three additional parameters, namely z 4, z 5 and V2. The ansatz of a third-order polynomial for V~(z, E) in the non-asymptotic region is of course an ad-hoc assumption and in fact one might devise different prescriptions in that region [28]. The barrier potential (2.29), (2.30) has been used extensively in LEED calculations by Rundgren and Malmstr~Sm [27]. It comprises the pure image potential with cut-off as a limiting case. Assuming the validity of (2.26) for the vector potential A one may calculate the surface contribution I~ in the one-step model for a general one-dimensional surface barrier V B. It is given by [29] IeB(kll, e/)

1 A0z ~. 2wclm[e~,'c',,B],

(2.31a)

where q = ((111, qz)=0 denotes the photon momentum and c~ = (call, caz) specifies the position of the surface barrier with respect to the first atomic layer. B has the form [29] B = E(V~_g + V ~ _ , ) e -ix2--, .... (a~g+ra,da-e+d~e) g

×

f?

dz' e '(q~+l(;.-*Az' ~(Kl-gz,(z'- clz )

-

)+ VB(z').

(2.31b)

Herein a,-g,Vi-g and afg, d3e are given in the notation of Pendry's original work [14] and denote the plane wave expansion coefficients in the constant

(7. Borstel, G. Th6rner/ Inversephotoemissionfrom solids

13

part z > clz of the barrier for the high- and low-energy electron wavefield, respectively, rtg is the internal low-energy reflection coefficient of V B and ~P denotes the z component of the low-energy electron wavefield in the non-constant part of the barrier. K2gz = - [ 2 ( E + co - Evac)

--

[--k[] q- g[

2] 1/2,

Kl-g~ = - [ 2 ( E - Vo - iVoi) - [ k f l - q l l - g[2] 1/2 For a step barrier at z = Clz one has dVS(z)/dz = (VoMT - Ev~¢)8(z - c1~) and with xO(K~-gz,0) = 1 one obtains from (2.31)

I~(k,,, el)

1Ao, ~r 2coc Im X(a~-s +

[(

v°MT-Evac) eiq'c~E(V~'-g+ V~_g)

rlsdl-s + d~-,)],

g

(2.32)

which corresponds to Pendry's result [14]. For the more realistic barrier potential (2.29), (2.30) the derivative dV~(z)/dz for the low-energy state, which enters (2.31), is known analytically. The same holds true for the z-component of the low-energy wavefield in the regions z < min(z 2, z4) and z > max(z 3, zs), where q~ may be formulated in terms of Whittaker functions and damped plane waves, respectively. In the intermediate polynomial range ~/" must be calculated numerically [27,29]. We recall that (2.31) is based on the acceleration form A . v V for the one-electron matrix elements. The validity of (2.26) for the vector potential A and its modification due to local-field corrections in the surface region has been investigated theoretically [30], but we will not discuss this point in more detail here. As has been mentioned already, the cross section for inverse photoemission is several orders of magnitude smaller than that for ordinary photoemission, and the concomitant experimental limitations thus will hardly allow for identifying local-field effects in inverse photoemission in the near future.

3. Inverse photoemission from bulk states Calculated inverse photoemission spectra for bulk states within the flamework of the one-step model have so far been reported by Th/Srner and Borstel for Cu(100) [31], by Jepsen et al. for Ni(111) [32], by Th6rner and Borstel for Ni(100) [33] and by Feder and Rodriguez for Fe(110) [34]. All these calculations utilized a step surface potential barrier, which is a quite reasonable approximation as long as surface states are disregarded. As typical examples

G. Borstel, G. Thfrner / Inverse photoemissionfrom solids

14

a

[u

(100)

~ :9

rXWK ••

%.

,-"

.,•'."

"°•.

• , ~.•" ............

••

• ••"

•

to -~_

. "'"'-

, - ""

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

•"%•° •

•••

" .°

Z

."

.

• ..".

%.•°

".

,•"'..

"..

••%

8 = 25*

8 = 23 ° °.°°.• ..............°°•'•°'•

-.

e

= 18 °

•'%.°..•..••••••••••.•,'-••.• •.

"'"

0 = 13 ° •••"•"• ....

"-..

•

"••'•-•.••.•.

•

.......

',

• •'

••.•....

;

e = 270 .... ...°•°. .....

••

•

.•

' ...........

•.

~•

.

••

8 = 28* ,.,.••.. .......

"'- .

°

j

,-

"'•

.'" .."'°.

•••

O = 30* "•-°••°•. ......

•

•° ,-i

7eV

j•'••••

i

i E-E F

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

;

(eV)

8=0~ ,-" ...........

i

;

o

~

2

3

~

5

{ E - E F)/eV Fig. 3. (a) Experimentally obtained inverse photoemission spectra (h ~o = 9.7 eV) for Cu(100) [35]. (b) Theoretical spectra calculated on the basis of the one-step model.

we present in figs. 3 and 4 a comparison between calculated and experimental angle-resolved inverse photoemission spectra for Cu(100) [35] and Ni(100) [331. The left panel in fig. 3 shows experimentally obtained inverse photoemission spectra for electron incidence in the F X W K plane of Cu(100) and photon energy h~o = 9.7 eV. The results of the one-step model calculation for kinematical parameters comparable to the experiment are given in the right panel. For the bulk muffin-tin potential of Cu the potential due to Chodorow [36] has been used. The step barrier (W = 4.59 eV) was at 0.8 interlayer distances outside the outermost atomic layer• The peak near 0.5 eV for 0 = 0 o must be attributed to a direct bulk transition A 1 --,A1. For off-normally incident electrons this peak disperses towards higher energies. The requirement that the initial state wavefunction must show even parity with respect to the F X W K mirror plane allows for an additional bulk transition, if 0 increases towards 29 o, but owing to final-state lifetime effects and instrumental broadening this

G. Borstel, G. ThSrner / Inverse photoemission from solids

I

t

I

I

I

15

I

Ni (001)[110]

-4--"

c

.0 >.Z iii I--Z

01

2345 ENERGY ABOVE

01

2345

E F (eV)

Fig. 4. (a) Experimentally obtained inverse photoemission spectra ( h ~ = 9.7 eV) for Ni(100). (b) Theoretical spectra calculated on the basis of the one-step model [33].

additional peak is not resolved from the main peak. The weak shoulder near 0.5 eV in the calculated spectrum for O = 29 ° stems from the truncation at the Fermi level E F and the subsequent folding with the experimental apparatus function. The excellent agreement between theory and experiment in fig. 3 is mainly a consequence of the Chodorow potential. This is not at all surprising since this potential has already been found to describe photoemission experiments on Cu with enormous success [37]. The overall agreement between theoretical and experimental inverse photoemission data for Ni(100) in fig. 4 looks also quite impressive. The spectra are calculated on the basis of a bandstructure potential for ferromagnetic Ni due to Moruzzi et al. [38]. The step surface barrier ( W = 5.07 eV) was at 0.8 interlayer distances outside the topmost atomic layer. Both majority and minority spin contribution have been added in the calculations. The strong emission enhancement between 1.8 and 3.6 eV is due to direct radiative bulk

G. Borstel, G. Thrrner / Inverse photoemission from solids

16

transitions from a primary-cone band and corresponds to the main emission feature in fig. 3. A second very weak dispersing structure appears for 0 > 19.2 ° between 1 and 2 eV and belongs to transitions from a secondary-cone band. The weak shoulder near E v in the calculated spectra is mainly due to truncation at E v and instrumental broadening. The well resolved peak near E F for large 0 in the experiment is not found in the calculations. Since both direct radiative transitions between bulk states and transitions from evanescent states into the flat d-bands of the minority spin system near E F are taken into account in the calculations, the experimentally observed emission feature near Ev has been attributed [33] to phonon-assisted non-direct transitions outside the scope of the calculations. Likewise it can be interpreted as an indication that the imaginary contribution to the effective bulk potential of - 0 . 7 5 eV for the upper unoccupied state is chosen too small in the calculations. In fact, Jepsen et al. [32] found in their work on N i ( l l l ) a value of - 1.3 eV and the concomitant increased relaxation of the perpendicular momentum conservation to result in a much more prominent minority spin d-band peak near E F. The fact that inverse photoemission from bulk states in Ni(100) is quantitatively described by a first-principles bandstructure potential must be regarded to a certain extent as fortuitous. Discrepancies in energy between theory and experiment, which are of the order of the experimental resolution (0.3 eV) for Ni(100) in fig. 4, increase to values near 1 eV for N i ( l l l ) [32,39]. Thus the failure of first-principles bandstructure calculations to account quantitatively for the occupied bands in Ni, as observed in photoemission, does essentially persist in the unoccupied band regime.

4. Inverse photoemission from surface states

4.1. Surface states in multiple scattering theory In this section we present general conditions for the occurrence of electronic surface states at clean surfaces in the framework of multiple scattering theory. It is based on arguments originally developed by Echenique and Pendry [40] and McRae [41]. We start with the low-energy wavefield ¢(rlt, z) in the region between the topmost atomic layer at z = 0 and the surface barrier, where the effective potential is spatially constant, cf. section 2.3:

¢p(rtl, z ) = ~ [ ( b ~ g e x p [ i K ~ ( r - c l ) ]

+b~gexp[iK~g(r-c~)]} ,

(4.1)

g

with Kl~=(kll-q,,-g,

+[2(E-

V0-iV0i ) -

Ik,,-q,,-g[2]l/:).

(4.2)

G. Borstel, G. Thrrner/ Inversephotoemissionfrom solids

17

This wavefield is a sum of two sets of plane waves, one set (big) headed towards, the other (b~-g) away from the barrier. According to Echenique and Pendry [40] and McRae [41] the surface state is viewed as a resonant condition of these two sets of plane waves. If the reflection coefficients of the crystal are denoted by rlgg" c and those of the barrier by r~g, we have

big = £ rlgg,blg+ gt

(4.3)

for the reflection by the crystal and

b;g = E r~g,ba-g, gt

(4.4)

for the reflection at the barrier. The requirement of self-consistency between (4.3) and (4.4) results in

det( I - rCraB) =O.

(4.5)

The resonance condition (4.5) can be further simplified by neglecting any rll-dependence of the barrier potential. The matrix r~ then becomes diagonal in g. For kit = 0 and energies near the Fermi level E F all beams with g ~ 0 in (4.1) are strongly decaying. Assuming that the multiple scattering via these strongly evanescent waves may be neglected (r~g = 0 for g 4= 0), we obtain from (4.5) 1

-

c

rloorlo

a o =_ 1 - rc exp(iq~c) r B exp(iq~B) = 0.

(4.6)

This condition can be fulfilled only if r c = r u = 1,

(4.7)

~c + ~B = 2~rn,

(4.8)

where n is a non-negative integer. Eq. (4.7) requires the absence of damping processes (Im V ~'r = Im V B = 0) and energies E in a relative bulk band gap (r c = 1) and below the vacuum level (r B = 1). This situation is schematically depicted in fig. 5. If E is outside these ranges or damping processes are taken into account, it is convenient to replace (4.6) by the weaker condition 11 - rc exp(i~bc) r B exp(iq~B) [ 2 ~ Min.

(4.9)

The solutions of (4.9) will then describe damped surface states inside a band gap and surface resonances outside the gap for klf = 0. The stationarity conditions (4.7) and (4.8) are well-known from the theory of bound states in one-dimensional quantized systems. The quantum number n in (4.8) has a precise mathematical meaning and gives just the number of nodes of the corresponding eigenfunction for finite z beyond the crystal edge. In the one-step model of inverse photoemission the full energy-dependent reflection matrices r c and r~ are calculated for the assumed potentials V m"

G. Borstel, G. ThiSrner / Inverse photoemission from solids

18

BARRIER

__BAND . . . . . . . . . . GAP

VACUUM LEVEL

L_

rae'*~

I

-

Irce

:"

-z

r-

i \': )

Fig. 5. Schematic potential in the vicinity of a crystal surface [42].

and V B, respectively. Minima in Idet(I-rCr~)l may then show up as maxima in the photon current. The rules for identifying a peak in an angle-resolved inverse photoemission spectrum as a surface state are exactly those known from ordinary photoemission [43]: (i) Energy E and parallel momentum kll of the state must lie in a gap of the projected bulk band structure; (ii) for fixed kll the energy E must be independent of the photon energy; (iii) the peak must occur predominantly for z-polarized light. The stationarity condition (4.8) has been used extensively by Smith and coworkers [7,42,44]. By assuming reasonable expressions for the energy-dependence of the phases ckB and ~c they have been able to give a qualitatively correct picture of surface states within the s-p band gap of various metal surfaces and to derive useful quantitative trends. Before presenting calculated inverse photoemission spectra for surface states, we will thus give in the next section a discussion of unoccupied surface states on C u ( l l l ) and Cu(100) within this simple one-beam (g = 0) multiple scattering model.

4.2. Examples." Cu(111)and Cu(lO0) In this section we apply the stationarity conditions for the existence of surface states derived in the last section to the s-p-like Shockley-inverted band gaps L 2, - L t in C u ( l l l ) and X 4, - X1 in Cu(lO0). We use various stages of approximations for the bulk potential and the surface barrier and study their effects on the electronic surface structure.

G. Borstel, G. Thrrner / Inverse photoemission from solids

19

For the L 2, - L 1 gap in C u ( l l l ) we use the phase condition (4.8). We consider the gap to be nearly-free-electron-like (NFE) and the corresponding surface states to be of the form described by Maue [45] and Goodwin [46]. For kij = 0 we have according to Smith [42] for the crystal phase ffc ~tan(½q~ c - z0~

) = ~

t a n ( 3 - z0 2 ~ o ) - "/.

(4.10)

Eq. (4.10) is written in atomic units with the energy E being referred to the inner potential F0 of the crystal, z 0 denotes the position of the crystal edge. E o is the energy difference between the center of the gap and V0. The phase 8 of the wavefunction exp(-3,z) cos(z 2 ~ - 3 ) inside the crystal z > 0 is given by sin 28

=

-'y2~o/Vo,

(4.11)

and the decay constant ~, by 3,2-- 2[(V2

+4EEo)'/2-(E + E o ) ] .

(4.12)

V~ is the Fourier coefficient of the bulk potential for the gap in question, i.e. 21 r e I is just the width of the gap. We take the origin z -- 0 at the center of a surface atom and have therefore Vo > 0 for a Shockley-inverted gap. From the bottom (L2,) to the top (L1) of the gap the phase 8 will then vary from - ~ r / 2 to zero, which in turn results in a p-like wavefunction sin(z 2 ~ o ) at L 2, and a s-like wavefunction cos(z 2 ~ o ) at L1. The model parameters E o and Vo are obtained from a regular band calculation. The abrupt cut-off of the bulk potential at z 0 can be chosen in the range of one interlayer distance d outside Z=0.

The phase ~B of a general barrier potential VB(z), whose non-constant part starts at z = z c and extends towards the negative z-direction, is given by ~pB = - - 2 Z c ~

- 2 arctan ~ - - - ~ Z c l

E)

'

(4.13)

where ~(z, E ) denotes the regular solution of the Schrrdinger equation for energy E inside the barrier. Eq. (4.13) follows from matching the (positive) logarithmic derivative of the wavefunction inside and outside the barrier. If E has its minimum value E = V0 = 0, we have q~a = --Tr independent of V B. If E takes its maximum value E = Evac, ~B depends on the nature of vB: For a short-range V B (Eva c - l T a ( z ) = I zl -k, k > 2, z ~ - oo) it will take some finite value q~B(Evac) > --~r/2, i.e. q~B remains bounded. For a long-range V B, however, ~a goes to plus infinity like ( E v a c - E ) -1/2 for k = 1 or like - l n ( E v , c - E ) for k = 2. The unboundedness of q~B for long-range barriers becomes crucial in the phase condition (4.8). Since q~c = 28 in (4.10) and 28 is bounded, eq. (4.8) can have only a finite number of solutions for a short-range barrier but has an infinite one for a long-range barrier potential.

G, Borstel, G. Thi~rner / lnoerse photoemission from solids

20

We recall that we have chosen the origin z = 0 at the center of an atom in the topmost atomic layer and the positive z-direction towards the crystal. This is the standard convention in one-step photoemission and L E E D calculations. Inverting the z-direction means changing ~, to - 7 and ~ to - ~ in the formulas above. 8 will then vary from ~r/2 to zero across the gap. Shifting the origin z = 0 to a position just half an interlayer distance outside the topmost layer means changing the condition VG > 0 for a Shockley-inverted gap to V~ < 0, and ~ will then vary from zero to ~r/2 across the gap. We further denote that some authors [42,47] omit in ~c and ~B the contributions 2z0 2v~E and - 2 Z c 2v~E-, which are accumulated in the free propagation, and introduce them as separate terms in the total phase balance. The most simple model for a barrier is an abrupt j u m p at z 0 = z c from the bulk potential to the vacuum level Evac. Such a step potential was originally studied by Maue [45], Goodwin [46] and Shockley [48] and gives the well-known result that the existence of a surface state in a N F E gap requires the inversion of the ordinary sequence s,p to p,s at the gap edges (Shockley-inverted gap). For realistic values of V~, E c , z 0 and Evac the phase condition (4.8) then allows only for the n = 0 solution, the Shockley surface state. The wavefunction of this state decays exponentially beyond the crystal edge z < z o and exhibits no node in that region. Since this surface state is a direct consequence of the termination of the crystal, it is called also crystal-induced [40]. All these conclusions remain essentially valid if one chooses for V B a general short-range barrier potential. Since in this case q'B remains bounded, it is in realistic situations not possible to get from eq. (4.8) more than the n = 0 solution for a N F E gap. A simple and analytically tractable long-range barrier potential is the image potential with cut-off at z c VB(z )

=

Evac + 0.25(z

vc

-

Zim) -1

(Zim > Zc > Z),

(4.14)

(z0>z>zc),

where Zim denotes the position of the image plane and Vc ---VB(zc). The vacuum level Ewe is given by the sum of the Fermi energy E F and the workfunction W. Vc is fixed by the value of the bulk potential at the cut-off point z 0 in (4.10). The regular solution of the Schr~Sdinger equation for V B ( z ) in the range z < z c, which enters (4.13), is the Whittaker function W is,1/2 (x e -i~) [49] with x = - 2 i k z ( z - zim), s = - 0 . 2 5 / k z, k Z-- [2(E - Eva~) I kll[2] 1/z. The phase condition (4.8) is solved graphically in fig. 6 for C u ( l l l ) assuming two different positions for Zim. The values for Vo, E o and E F are taken from the band calculation of Burdick [50], which is based on the Chodorow potential ]36], and W = 4.98 eV [51]. The results of such a phase analysis have been shown to be not very sensitive to the choice of the crystal edge z 0 [47]. We have chosen z 0 = - d / 4 , i.e. the bulk potential is terminated

21

G. Borstel, G, Thi~rner / Inverse photoemission from solids

-TI;

0

re

2n;

I

I

I

3re

I

I

VACUUM LEVEL

> OJ

L1

>LU

Z LLI Z O rr I-O LLJ ._J LLI

-- a:Z,m=_0.52~ 2 -- b:zim =-0.82A

0

na:[ i

,n~:O

k,. Cu (111)

-2

/A - T'C

0

t

I

rc

2re

3r~

Fig. 6. Energy-dependence of the phases ~c and ~a in the multiple reflection theory of Cu(lll) for an image potential with cut-off and image plane at position zim. Solid circles denote the energies of the n = 0,1 surface states. at the inner potential V0 --- 0, which means Vc = 0 and z c = 2im 0.25/Eva c in (4.14). Fig. 6 shows that one obtains again the n = 0 (crystal-induced) Shockley surface state, k n o w n already f r o m the simple step barrier. Physically this arises from the fact that the Shockley state is due to the termination of the crystal and thus its existence independent of the nature of V B. Since @B n o w diverges as E approaches the v a c u u m level, the phase condition (4.8) results in a series of additional states n >__1 just below Evac. These states are a direct consequence of the image behaviour of V ~ and therefore have been termed image-potential surface states [52]. The physical nature of this new class of surface states can be obtained by reference to standard atomic physics: F o r a hydrogen-like atomic system with nuclear charge Z the binding energy of the n t h level is given by - 0 . 5 Z 2 / n 2. F o r this atomic system the radial SchrSdinger equation for s-states and the corresponding b o u n d a r y conditions are formally the same as those which one obtains for the one-dimensional pure image potential. Transferring this result to the image potential where Z = 1 / 4 , we therefore expect for the energies E n of the image-potential states E n = Evac 1 / 3 2 n z. The saturation of the image potential, i.e. the s m o o t h transition to the bulk potential as a result of m a n y - b o d y screening processes, leads to correc-

-

22

G. Borstel, G. ThOrner / Inverse photoemission from solids

tions in this hydrogenic behaviour, which can be taken into account b y introducing a q u a n t u m defect an: En = Evac - 1 / 3 2 ( n + a n

):

(n = 1, 2, 3 . . . . ).

(4.15)

In the hydrogenic limit a n = 0 the first three states occur at binding energies - 0 . 8 5 , - 0 . 2 1 , - 0 . 0 9 eV relative to Eva c. Since typical workfunctions W are of the order of some eV, the image-potential states are usually unoccupied and their detection requires techniques like inverse photoemission. Outside the gap, where r c < 1, one cannot apply the phase condition (4.8) in a strict sense but must resort to the more general condition (4.9). If Eva c is above the upper gap edge, as it is the case for C u ( l l l ) in fig. 6, the divergence of ¢~ on approaching Eva c overrides the energy-dependence of r o and r c ( E ) in that range m a y be a p p r o x i m a t e d by some average value between unity and rc(Evac). For constant r c the phase condition (4.8) will, however, hold again, i.e. it will locate image-potential states also above the u p p e r gap edge. Exactly this situation of image-potential surface resonances is encountered in fig. 6. The error from approximating r c ( E ) in that region by a constant is smaller than the uncertainty due to the arbitrariness in choosing the cut-off point z 0 in (4.10). The typical behaviour of the surface state wavefunctions '/'n ( z ) is shown in fig. 7. Beyond the crystal edge z < z 0 the n = 0 crystal-induced surface state exhibits an exponentially decaying tail without any nodes, just as the n = 0 Shockley surface state for a simple step barrier. I q'012 has its m a x i m u m near z0, i.e. slight variations of the effective potential near the surface will have a

I

I

I

1

I

I

I

I

~n(z ) n=0

n=l

n=2

h

30

I

J

JI

0

J

i

i

-30

l

I

-60 zl~) Fig. 7. Surface state wavefunctions for an image-potential barrier [1].

G. Borstel, G. ThOrner/ Inverse photoemission from sofids

23

-2

>

b

-&

a

-10 -12 -IZ -4

i -3

1 -2 z

Fig. 8.

I -I

0

(i)

Two image-potential barriers for Cu(lll) resulting with 0.0l eV in the same series of surfaces states n > 0 a t kll = 0.

quite large effect on the energetic position of the n = 0 state. By contrast the n = 1,2 image-potential states with one and two nodes for z < z 0, respectively, protrude far out into the vacuum region and are therefore only slightly affected by the bulk potential. As a variation of the effective crystal potential causes a change in the workfunction, this means that the image-potential states are essentially tied to the vacuum level. According to fig. 6 the energetic position of both the n = 0 and the n = 1 surface state on C u ( l l l ) depends on the assumed position Zim of the image plane. Since both states have been measured by ordinary photoemission (n = 0) [53], inverse photoemission (n = 1) [54-56] and two-photon photoemission (n = 1) [57,58], one might try to extract Zim from these experimental data by analysing the image potential with cut-off. This has recently been done [47,59], but the extracted values for zi~ should be regarded with caution, since even for fixed q~c(E) the extracted position of the image plane depends strongly on fine-details of the assumed barrier. This is demonstrated in fig. 8, where two image-potential barriers for C u ( l l l ) are shown, which within an accuracy of 0.01 eV result in the same surface states n > 0 at kll = 0. Barrier (a) is an image potential with cut-off (Zim = --0.82 A), the results of which are also shown in fig. 6. Barrier (b) is a smoothly saturating image potential according to eq. (2.29) with Zim = - - 0 . 6 5 A. The reason for this strong ambiguity lies in the fact that fiB(E) must be obtained by integrating the Schr~Sdinger equation, i.e. q~B(E) is an integral property of V R ( z ) . If one considers only surface states at kll = 0, the experimental input information is typically restricted to the energy range between E F and Evac. The barrier V B,

24

G. Borstel, G. Th5rner / Inverse photoemission from solids

however, extends in energy from Vo the Evac and it is then immediately clear that additional experimental information between V0 and E F is needed to arrive at a more conclusive V B. Obviously this additional input will come from experiments at kll #: 0, since a surface state at E(kll ) probes the barrier V B at the energy E z = E(kll ) - kl~/2. For a consistent description of surface states at kLi ~ 0 within multiple scattering theory one must restart with (4.5) and take care to include correctly all propagating plane waves in the basic expansion (4.1). According to (4.2) one has to take into account all beams for which I kll-g12_< 2(Eva c - V 0 ) ( = 2 5 eV for C u ( l l l ) ) . A continuous transition from k l l = 0 (F) to the boundary of the surface Brillouin zone requires at least two-dimensional matrices r c and rfl in (4.5). If such a two-beam description is possible and kll :~ 0 coincides with a point of high symmetry, one can split eq. (4.5) into two one-dimensional equations and may maintain for each of them the phase condition (4.8) [42]. We now turn to the X 4, - X 1 gap in Cu(100) and consider again kll = 0. When compared with the L 2, - L 1 gap, the X 4, - X 1 gap occurs at higher energies. The vacuum level falls near the center of the gap, resulting in image-potential states inside and a crystal-induced surface state below the gap. Because the barrier phase e#B(E) exhibits no singular behaviour below the lower gap edge, the energy dependence of rc becomes non-negligible there. This means that the phase condition (4.8) is no more valid for the crystalinduced surface state at Cu(100) and one must resort to (4.9). If the N F E approximation is discarded and both e#c(E) and r c ( E ) are calculated for Cu(100) from the Chodorow potential [36], one obtains the results shown in fig. 9. To mimic damping effects due to the imaginary part of the self-energy, a small imaginary contribution V0i = - 0 . 0 5 eV has been added to the inner potential V0. Its main effect is to smooth out discontinuities in d r c / d E and de#c/dE at the gap edges which would appear for t h e u n d a m p e d case. The first two solutions of eq. (4.9) for the Chodorow potential and the imagepotential barrier with cut-off (Zim = - 0 . 7 4 ~,, W = 4.59 eV [51]) are shown as n = 0 and n = 1 in fig. 10. The n = 1 image-potential surface state is clearly identified by the sharp minimum near 4 eV. At the vertical scale of fig. 10 this minimum is indistinguishable from zero and its energetic position could likewise be obtained from (4.8). By contrast the crystal-induced surface state near 1.15 eV appears as an extreme weak minimum which can be identified only at an enlarged vertical scale (see insert in fig. 10). Note that at 1.15 eV the reflection coefficient rc has already dropped to - 0 . 4 . The crystal-induced surface resonance at Cu(100), which has been experimentally detected at 1.1-1.15 eV [60,61], thus exhibits a large bulk component and for its location the correct energy-dependence of r c ( E ), e#c(E) and e#B(E) becomes crucial in eq. (4.9). We have seen that the simple one-beam phase model in the two-band N F E

25

G. Borstel, G. Thi~rner / Inverse photoemission from solids

1.0

0

Cu(100) k,=0

¢c

Cu(100) k , , =

rC

(a)

(b)

0.8

-1

0.6 0.4

-2

X/, I

0.2-3

0

I

I 1

,

I 2

J

I 3

I

I 4

0,0

J

0

I

J

1

E- EF (eV)

I

~

I

2 3 E- E F (eV)

I

I

Z,

Fig. 9. Energy-dependence of the phase ~ (a) and the modulus r c (b) of the reflection coefficient for Cu(100) near the lower gap edge X4,.

a p p r o x i m a t i o n gives a q u a l i t a t i v e l y correct a c c o u n t of the physics o f surface states in the u n o c c u p i e d s - p gaps in metals. F o r d e t a i l e d q u a n t i t a t i v e investig a t i o n s in the surface Brillouin zone, which are n e e d e d for a m o r e t h o r o u g h investigation of the effective b a r r i e r p o t e n t i a l , it is clearly o v e r c h a r g e d a n d we

Cu (100) k,=0 / / / ~

°io ~2

~t,

1

00

1

1

2

3

z,

E-E F (eV) Fig. 10. Energy-dependence of I1 - rcexp(i~c)raexp(i~B)[ 2 in eq. (4.9) for Cu(100) and an image-potential barrier with cut-off (Zim= -0.74 ,~) near the lower gap edge X 4',

26

G. Borstel, {7, Thrrner / Inverse photoemission from solids

therefore will come back in the next section to the more rigorous three-dimensional approach.

4.3. Inverse photoemission and effective surface barrier in Cu(lO0) The multiple scattering theory combined with N F E theory in the preceding section provided a qualitatively correct insight into the systematics of surface state formation in s-p-like band gaps of metals. In what follows we will concentrate on more quantitative aspects. Assuming that a reliable bulk potential V MT is at hand, one may ask what information on the precise form of the surface barrier potential V B can be derived from inverse photoemission experiments on unoccupied surface states. In contrast to the asymptotic z-1-behaviour of V B, which already follows from elementary electrostatics of macroscopic media, the calculation of the position Zim of the image plane poses a difficult many-body problem, which even for simple systems has not been solved exactly. For a static uniform background (jellium) model of a semi-infinite metal Zim can be related to surface properties, since it coincides with the center of mass of the induced surface charge relative to the edge of the positive background [62]. The position of the image plane Zim depends on the method of calculation and has so far been found to lie 0.5-1.6 a.u. outside the jellium edge for metallic densities (r s = 2-5) [63]. The jellium edge is, by construction, at half an interlayer distance outside the topmost layer. For an electron gas parameter r s = 2 and d/2 = 0.9 ,~, which are appropriate for Cu(100), one expects the position of the image plane at Zim = -- 1.6 A outside the outermost layer. Compared to this result of jellium theory the positions of Zirn, as used in the preceding section, are significantly shifted towards the topmost layer. If this trend could be verified in a more rigorous calculation, it would clearly be of high significance for the theory of semi-infinite metallic systems. For such an investigation Cu(100) is ideally suited, since the Chodorow potential [36] is known to give an excellent description of bulk properties in Cu and since there exists now a sufficiently large body of experimental information on empty surface states at this face. For Cu(100) so far four empty surface states SR, Sa, $2, $3 and two unoccupied bulk states B 1, B2 have been detected in inverse photoemission experiments along the F X U L azimuth. The experimental energy versus parallel m o m e n t u m dispersion E(k,) of these states is shown as solid dots in fig. 11. SR denotes a crystal-induced surface resonance [60,61] and S 1 the first image-potential state [64,65], both being associated with the X4, - X1 gap. S2 and S 3 represent crystal-induced surface states derived from the L 2, - L 1 gap [64,65]. S3 becomes an occupied state near ,X and has been detected there in high-resolution photoemission spectroscopy [66] (solid triangle in fig. 11). So far information on the effective barrier at metal surfaces has come to us

G. Borstel, G. ThOrner / Inverse photoemission from solids

27

Cu(100) F'XUL

87 --

6 ,

231

///~X

° B2/,

$2~

OP Fig. 11. Bulk bandstructure of Cu, projected onto the F - X direction of the surface Brillouin zone. Solid circles and triangles denote experimental data. Heavy solid lines result from calculations employing the one-step model of inverse photoemission [67].

primarily from the comparison of experimental and calculated fine-structure in LEED data. Typical results from such a LEED analysis for Cu(100) are shown in fig. 12a. According to Read [68] two modified (MIB) and one saturated image barrier (SIB) account within 0.05 eV for the Cu(100) high-resolution data of Dietz et al. [69]. The position Zirn of the image planes in fig. 12a is determined within 0.2-0.3 a.u. at zim = - 8 . 6 a.u., zim = - 5 . 2 a.u. (MIB) and Zim = - - 3 . 7 a.u. (SIB). Although the barriers of type MIB in fig. 12a are certainly unphysical, since they would give rise to a huge dipole barrier due to a dramatic flow of electrons into the vacuum region, these results show clearly that the problem of determining the detailed barrier shape for Cu(100) from LEED data is essentially unsolved. In order to derive a more definite barrier potential for Cu(100) one may start with the quite general barrier potential V B in eqs. (2.29), (2.30), calculate inverse photoemission spectra for the various surface states in fig. 11 and vary V a until one obtains coincidence in energy for all four surface states within the reported experimental accuracy. This has been done recently [67] and the results will be discussed in the following in more detail. The experimental data for empty surface states at Cu(100) shown in fig. 11 have been recorded by using an acceptance angle of 28 ° for the photon

28

G. Borstel, G. Th6rner / lnverse photoemission from solids i

-2

1

I

I

I

,

,

i

I

I

t

~tIE

-6 (o)

A

>O3 -10 1

1

1

i

i

ill

J

cn i

I

rr® --62 Cu(lool -10

(b) -15

-10

-5 z (o.u.)

Fig. 12. (a) Different barrier potentials of Cu(100) used for analysis of LEED fine-structure data [68]. (b) Barrier potential from analysis of inverse photoemission, photoemission and two-photon photoemission data. The origin z = 0 is at the center of the first row of atoms [67].

counter [64,65]. The concomitant averaging over different directions of the optical vector potential A prevents to derive reliable quantitative results from the measured intensities. This means that details of Im V a cannot be obtained. Even for more selective experimental geometries the uncertainty with regard to the influence of local-field corrections in A near the surface would persist and make quantitative results for I m V B unreliable. For energies E near the Fermi level, however, one clearly has IIm V B [ << I E - V0 I and according (2.10) the energetic positions of the empty surface states in fig. 11 will therefore be independent of details in I m V B. The main goal is thus to determine the real part Re V B of the effective barrier potential in (2.29). The exchange contribution V l ( Z - Zim)-2 in (2.29) is expected to represent a small correction to the correlation term 0.25 ( z - Zim) -1 [25] and its influence is therefore absorbed into the polynomial range between z 2 and z 3. This reduces the number of parameters in (2.29) to three, namely Zim, Z2 and z 3. Since the empty surface states occur near EF, where V0(E ) according to fig. 1 is only slowly varying, the inner potential for the final state may be regarded as a constant, V o ( E ) = Vo(EF)= Vo. The initial state of the inverse photoemission process is at 10-15 eV above E v ( h ~ = 9.7 eV) in the present case. The reflectivity of the surface barrier in that energy region has already dropped to values less than 0.1 and one therefore can work for the initial state in the classical limit r ~ = 0. Both inverse

G. Borstel, G. Th&ner / Inverse photoemission from solids

29

photoemission [9] and LEED [70,71] studies for Cu indicate that Vo(E ) is only weakly energy-dependent in this energy range, i.e. Vo(E) may be taken as Vo(Ev) -- V0 also for the initial state. From the experimental data shown as solid dots and triangles in fig. 11 and additional high-resolution data for the n = 1,2 image-potential surface states SI(F), S~(F) as determined by two-photon photoemission [72], one arrives at the result in fig. 12b. It shows two (unresolved) extremal curves for the Cu(100) barrier which are still compatible with the experimental error bars for the surface states SR, S 1, S2, $3 in figo. 11. From the average of these two extremal curves one gets Zim = - - 1 . 0 2 A, z 2 = - - 2 . 1 ,~, z 3 = - 0 . 0 0 2 ,~ for a workfunction W = 4.63 eV [72] ( E v a c - V0 = 12.2 eV) measured in situ. The solid lines in fig. 11 are calculated, using this optimal barrier potential. It is seen that the agreement between theory and experiment for the true surface states $1, S2, S3 is excellent. The energetic position of the surface resonance SR is difficult to measure [60,61], for the present optimal barrier SR has a large bulk component, as expected already from fig. 10, and is therefore not very sensitive to details of V B. Slight differences between theory and experiment for SR, B 1, B2 in fig. 11 must therefore be attributed to the bulk potential [36] of Cu. Fig. 11 also shows the calculated n - - 2 image-potential state (S~) whose dispersion has not been measured so far. A detailed comparison between experiment and theory for all observed surface states at F and is given in table 1. With the exception of the energetic position of SR(F), which, however, is to a large part a bulk property [73], and the effective mass m * of S3(X) all calculated results are within the error bars of the experiment. The slight discrepancy in m*(S3) is demonstrated in fig. 13, which shows the dispersion of the occupied crystal-induced surface state S 3 as calculated in ordinary photoemission ( h ~ = 11.85 eV) for the optimal barrier. This dis-

Table 1 C o m p a r i s o n of e x p e r i m e n t a l and c a l c u l a t e d b i n d i n g energies E a a n d effective m a s s e s m * of surface states on Cu(100); b i n d i n g energies for SR, S 2 a n d S 3 are given relative to the F e r m i level E F, b i n d i n g energies for Sl, S~ a n d S~' relative to the v a c u u m level Evac, respectively Type

E~ xp (eW)

E~he°r (eV)

SR S 1 (n = 1 ) S~ (n = 2) S~' (n = 3)

- 1 . 1 ± 0 . 2 a) - 0 . 5 7 : t : 0 . 0 3 b) - 0 . 1 8 ± 0 . 0 3 b) -

-1.5 --0.54 --0.17 -0.046

,X

Sz S3

3.6 -I-0.2 c) - 0.058 ± 0.005 d)

3.58+0.03 - 0.063

a) Ref. [60].

b) Ref. [72].

c) Ref. [65].

Symmetry point

d) Ref. [66].

m exp*

m theor*

-

0.9__+0.1 bJ

0 . 7 ± 0 . 2 c) 0.067 ± 0.01 d)

0.4 0.993 _+0.002 1.010 + 0.002 1.026 + 0.002 0.77 _+0.04 0.035 ± 0.005

30

G. Borstel, G. Th6rner / Inverse photoemission from solids

$3 -20

/

-.40 E kl_

,,iI -60 w

Cu (100) hw=11,85e~

-80

-100

~

-006

t

P

x

J

-002 0 002 k,-1231 (A-l)

t

006

Fig. 13. Bulk bandstructure of Cu, projected onto the F - X direction of the surface Brillouin zone, near X. Solid circles represent experimental photoemission data [66], solid squares result from calculations on the basis of the one-step model. The energy resolution of the experiment is indicated at ,X.

crepancy may be an indication that for energies Ez = E(kij) - kill2 near V0, as it is the case for S3(X), corrugation in the effective barrier potential might play a role. Comparing the optimal barrier in fig. 12b with the results derived from L E E D calculations, one clearly recognizes the progress which is made by inverse photoemission spectroscopy. The main reason for this success is that inverse photoemission detects empty surface states bound to the effective barrier and thus probes V B directly, whereas in L E E D the desired information on V B must be derived indirectly from the observation of the high-lying scattering states. A reliable determination of the barrier shape requires experimental data for surface states with energies E z covering most of the energy range of the barrier from V0 to Evac. In addition the experimental energy resolution must be sufficient. For example, the optimal barrier in fig. 12b could not have been obtained so precisely without the high energy resolution of 0.03 and 0.005 eV for SZ(F) [72] and S3(X ) [66], respectively. An energy resolution of 0.3 eV for SI(F) will result in an uncertainty of order 0.8 A for Zim and studies, which are based on such experimental resolution, can therefore not be regarded as conclusive with respect to the precise position of the image plane.

G. Borstel, G. Thi~rner / Inverse photoemission from solids l

I

i

I

i

i

30

~o

31

i

Cu {lOO) 'Bw=9.7eV

I \

so

SR S,1

_d

>I.u3 Z LLI I--Z

~heor

theor vo[uted~

I--I

/

J

I

0

I

2

~

i

i

4

I

6

E - E F (eV) Fig. 14. Comparison between calculated and experimental inverse photoernission spectra for Cu(100). The insert shows experimental data for the n = 1 image-potential surface state (solid circles) recorded with improved energy resolution [74] and the corresponding theoretical result (solid line).

A comparison between experimental and theoretical inverse photoemission spectra for Cu(100) is shown in fig. 14. The spectra are calculated for normal incidence (0 = 0 °) of the electrons and a photon energy h~0 = 9.7 eV, using the optimal surface barrier potential of fig. 12b. The raw data, obtained for light polarization along x, y and z, have been added for an average detection angle a = 37 ° to the surface normal [65] (upper theoretical curve in fig. 14) and convoluted with the experimental apparatus function (lower theoretical curve in fig. 14). The agreement between theory and experiment is excellent for the energetic positions of the peaks and quite satisfying for the relative intensities. This may be seen also from the insert in fig. 14, which shows the series of image-potential states recorded with increased experimental resolution [74] (solid dots) and the corresponding calculated spectrum (solid line), convoluted with the new apparatus function. For a one-dimensional effective surface barrier Va(z) surface states should show up predominantly for z-polarized light, i.e. they should become suppressed by decreasing the average detection angle ct relative to the surface

32

G. Borstel, G. ThSrner / Inverse photoemission from solids I I I I i I I I Cu {100) I"XUL cq=137°-81 h~=9 7eV

|

i

I

i

I

I

I

I

~2= 9 0 ° - 8 j\

a

i'!

b

/, t

i

B1 $2 82

, .~

_

2

$

i I

~'zL' I *-'1 I I I I I I I 0 2 4 6 8

0

[111111 2 L 6 8

E - E F (eV) Fig. 15. Comparison between calculated (solid lines) and experimental (dashed lines) [65] inverse photoemission spectra for Cu(100). 0 denotes the angle of electron incidence and c~ the average photon detection angle, both with respect to the surface normal.

normal. This effect is demonstrated in fig. 15. In order to prevent a loss in information due to the convolution procedure the experimental data are c o m p a r e d with the calculated raw spectra. The suppression of $2 for small c~ is clearly seen in fig. 15. As a further interesting detail fig. 15 shows a narrowing of the bulk peak B 1 for increasing angle 0 of electron incidence. This happens, since for the 0-values in fig. 15 the dispersion curve E(kll ) of B] just crosses the b o u n d a r y between bulk states and the L 2, - L a derived b a n d gap (see fig. 11) and becomes the dispersion of the crystal-induced surface state S 3. The agreement between experiment and theory for the relative intensities in fig. 15 is not as satisfying as in fig. 14, indicating that the superposition of partial spectra for x, y, z-polarization according to an average p h o t o n detection angle a is a crude approximation to the experimental situation. F r o m a theoretical point of view a reduction of the acceptance angle of the p h o t o n counter is therefore highly desirable. We have seen that the one-step model of inverse photoemission c o m b i n e d with experimental data on empty surface states allows for deriving a one-di-

G. Borstel, G. Thrrner / Inverse photoemission from solids

33

mensional effective barrier potential for metals with high accuracy. In particular the position of the image plane can be determined and has been found to be significantly shifted towards the topmost atomic layer, when compared with the predictions of standard jellium theory [62,63]. The theoretical explanation of this point is controversial at present [75,76]. It should be denoted that the position of the image plane is crucial in calculations of the tunnel current across the vacuum gap in scanning tunneling microscopy [77] and this fact further motivates the large interest in this quantity. 4. 4. Effective masses of image-potential surface states The calculated effective masses m n* of the n = 1,2,3 image-potential surface states on Cu(100) are essentially unity, cf. table 1, and this is exactly what one expects in view of the fact that the maximum probability I g'n(z) J z for these states occurs several Angstr~3m units outside the crystal edge, see fig. 7. There exist, however, experimental data on image-potential surface states at other samples, which indicate that m* may be significantly above unity [56], and this effective mass enhancement has been the subject of several theoretical investigations recently. A first discussion of effective masses associated with image-potential states has relied on surface corrugation. Garcia et al. [78] derived from a truncated second-order perturbation calculation a relation between effective masses mn and energy corrections A E, due to surface corrugation which reads (corrected for a factor of 4 error in ref. [78]) AE, = g2(1 - l / m * ) / 4 ,

(4.16)

where g denotes the modulus of the smallest nonzero surface reciprocal lattice vector. The inability of this relation to explain the observed mn* may be shown by reference to experimental data for Ni(111) [79]. With g - 1 . 5 7 a.u. and m~' = 1.6 one arrives from (4.16) at the strange result that A E 1 = 6.3 eV is comparable to the bulk L-gap of 7 eV which supports the surface states. Since Pendry et al. [80] have shown by explicit inverse photoemission calculations that corrugation effects in silver have no effect on the binding energies or on the effective masses of image-potential states and Giesen et al. [58] did not find the correlation (4.16) between binding energy and effective mass in their experiments, surface corrugation may definitively be ruled out as the source of the observed effective mass enhancement. In a second attempt to explain effective masses different from unity one may look more carefully at the energy-dependence of the effective surface barrier potential VB(z, E). This has been done by Echenique and Pendry [81,82], who found that the increase of the effective mass of image-potential states can be at most 2% due to plasmon-related m a n y - b o d y effects and 2% due to electron-hole pair excitations. This negative result is somehow disap-

34

G. Borstel, G. Th6rner / Inverse photoemission from solids

pointing, since in the case of an affirmative result image-potential states at metal surfaces would represent a well-defined two-dimensional electron system to study dynamical m a n y - b o d y effects experimentally. O n the other hand, this result backs the neglect of dynamical corrections in Re V ~, which has been made in our calculations for Cu(100) in the preceding section, and therefore facilitates the extraction of an effective barrier potential from inverse photoemission data. The correct explanation for the increased effective masses of image-potential states turns out to be quite simple. The effective mass is experimentally determined by measuring the energy of the surface state for at least two different values of kll and by subsequent fitting of the experimental data to a parabola E,(kll ) = E , ( F ) + k l l2/ 2 m , , . For increasing kth the energy En(kll ) will, however, move closer to some bulk gap edge, cf. fig. 11. On a p p r o a c h i n g a gap edge at E e the decay length of a b o u n d surface state diverges, i.e. the corresponding wavefunction will probe more and more bulk layers and in the u n d a m p e d limit will be spread over the whole macroscopic crystal for E = E e. This means that on approaching a bulk gap edge effective masses considerably different from unity must be expected which in addition will be kiFdependent. This kll-dependence renders a comparison between experiment and theory more difficult, as most of the experimental data on m n so far represent an average over some range of m*(kll) values [56]. We prove this effect by a simple model calculation of inverse photoemission from image-potential states on C u ( l l l ) . Using the C h o d o r o w potential [36] for the bulk crystal and an image-barrier potential with cut-off at the inner potential V0 we calculate the energy E a of the n = 1 image-potential state as a function of the angle 0 of electron incidence for hco = 9.7 eV, W = 4.98 eV. The results E 1 - E F are shown in table 2a for different assumed positions Z~m relative to the topmost atomic layer. Changing Zim allows a shift of the energy of the n = 1 state at F (0 = 0, kll = 0) across the L 2, - L~ gap and the resulting effect on m~' to be studied. The last two columns in table 2a give the effective mass m~' as calculated for the pair of values E1(F ) and El(O) from m* = [ E ( 0 ) - E v + c o - W ] sin20/[ E ( O) - E ( F ) ] . The results in table 2a clearly exhibit the anticipated trends. If the n = 1 state at F lies close to the upper gap edge L 1 ( E ( L 1 ) - E v = 3.98 eV), as it is the case for Zim = - 1.4 A, rn~' is increased since the effective mass rn* of the upper gap edge in Cu(111) is larger than unity. If El(H ) moves away from the gap edge, m~' decreases towards unity (Zim = - 1 . 8 A) and eventually becomes even less than unity (Zim = --2.2 ,~) due to the fact that El(kll ) now will approach the lower gap edge in C u ( l l l ) , which exhibits m* < 1. The kkl-dependence of m*1o is large near gap edges, see the last two columns in table 2a for Z~m = --1.4 A, and negligible well inside the gap. The behaviour of m , as a function of n is presented in table 2b for

35

G. Borstel, G. Th6rner / Inverse photoemission f r o m solids

Table 2 Calculated binding energies relative to the Fermi level E v and effective masses m* as a function of the angle 0 of electron incidence for the image-potential states on Cu(lll). For the surface barrier an image potential with cut-off has been used. The position of the image plane 2im is given relative to the topmost atomic layer (a) Binding energies E ] - E v and effective masses ml* for the n = 1 image-potential state for various positions of the image plane

Zim (m) 1.4 - 1.8 -2.2 -

E] - E v (eV) 0o 9.8 o

20 o

0 °/9.8 o

0 °/20 o

3.71 2.85 1.67

4.35 3.67 2.58

1.5 1.2 0.9

1.7 1.2 0.9

3.88 3.03 1.87

m~'

(b) Binding energies E n - E v and effective masses m* for the first three image-potential states on Cu(lll) for a fixed position of the image plane Zim = -1.4 ~,

n =1 n=2 n= 3

E. - E F (eV) 0o 9.8 ° 3.71 4.61 4.84

3.88 4.86 5.11

20 o

m* 0 °/9.8 o

0 °/20 o

4.35 5.76 6.07

1.5 1.1 1.0

1.7 1.1 1.0

Zim = - 1 . 4 A. It is seen t h a t in the case at h a n d m n* g o e s to u n i t y w i t h i n c r e a s i n g n. T h i s o c c u r s since for l a r g e r n the m a x i m a o f I q ' . ( z ) l 2 m o v e more and more inside the vacuum region and the concomitant decoupling of t h e states f r o m the b u l k crystal is the d o m i n a n t e f f e c t in this case. T h a t s o m e t i m e s the s i t u a t i o n c a n b e m o r e c o m p l i c a t e d m a y b e seen f r o m the last c o l u m n in t a b l e 1: F o r C u ( 1 0 0 ) the c a l c u l a t e d e f f e c t i v e m a s s e s o f i m a g e - p o t e n tial states are e s s e n t i a l l y unity, b u t the v e r y w e a k i n c r e a s e o f m n* w i t h n r e f l e c t s a r e s i d u a l b u l k e f f e c t w h i c h in this c a s e s l i g h t l y o v e r c o m p e n s a t e s t h e decoupling mechanism. A similar but more heuristic view of effective masses of image-potential states has b e e n g i v e n r e c e n t l y b y G i e s e n et al. [83] o n t h e basis o f t w o - b a n d N F E t h e o r y a n d the o n e - b e a m p h a s e m o d e l a p p l i e d to kLi ~ 0. If ktL is n o t t o o far f r o m F, o n e m a y m a i n t a i n t h e p h a s e r e l a t i o n (4.8) a n d the set o f e q u a t i o n s (4.10)-(4.13) with E(F) replaced by E z = E(kll ) -kill2. Without additional a s s u m p t i o n s this s u b s t i t u t i o n results in m e* = m ,* = 1, since in ( 4 . 1 0 ) - ( 4 . 1 2 ) o n l y o n e b u l k F o u r i e r c o e f f i c i e n t VG w i t h G n o r m a l to the s u r f a c e is t a k e n i n t o a c c o u n t . But if o n e f u r t h e r a s s u m e s t h a t t h e net e f f e c t o f all o t h e r b u l k F o u r i e r c o e f f i c i e n t s m a y b e s i m u l a t e d b y a k l ? d e p e n d e n c e o f the e f f e c t i v e g a p w i d t h 2 1VG I a n d the e f f e c t i v e g a p c e n t e r E ~ , this b u l k e f f e c t s h o w s u p as a k l l - d e p e n d e n c e o f t h e crystal p h a s e 4'c in (4.10), w h i c h in t u r n v i a the p h a s e

36

G. Borstel, G. ThOrner / Inverse photoernission from solids

relation (4.8) may then result in effective masses different from unity. V~ (kll) and E ~ ( k j j ) can be taken from a regular bulk band calculation. Within the limitations of two-band N F E theory this approach is valid near F and the findings agree qualitatively with the results of our calculations in table 2. 4.5. Spin-split surface states at f e r r o m a g n e t s

The multiple scattering approach for surface states, which has been discussed in section 4.1, can also be applied to ferromagnetic materials. In the spirit of density functional theory band ferromagnets are characterized by two subsystems, namely the minority and the majority spin system. One then can apply the multiple scattering theory to both spin systems separately. This will be demonstrated for the F e ( l l 0 ) surface. The bandstructure for Fe [38] near the N-point, modulus r c and the phase 4c of the reflection coefficient for the semi-infinite crystal are shown in fig. 16 for both spin systems. The calculations are done for kll --- 0 and include a small imaginary contribution V0i = - 0 . 0 5 eV to the spin-dependent effective bulk potential, which describes damping effects. It is seen that both spin systems exhibit an "effective" gap, where rc --1. In the minority spin system this effective gap extends from N 4 (1.5 eV) to N 1 (9.36 eV), and in the majority spin system from N3,N 1, (0.4 eV) to N 1 (7.78 eV). The increase of r c between N 1 and N1, for the minority spin system is due to the fact, that a plane wave in the vacuum region for klh = 0 can couple only to a state with Y~a symmetry in the bulk, which is not present in that energy region. The crystal phases 4(extend for both spin systems from approximately -~r (lower gap edge) to approximately - ~ r / 2 (upper gap edge), but are shifted against each other because of the spin-dependence of the effective bulk muffin-tin potential. The quite different behaviour of the crystal phases, when compared to that in the two-band N F E model, is predominantly due to the strong s - d hybridization in Fe and to a minor extent due to damping effects. To investigate qualitatively the possibility of spin-split surface states in ferromagnets, it is convenient to restrict oneself to the energy region, where rc = 1 for both spin systems. In the asymptotic limit z ~ - ~c, the z - 1 image-potential term, which is due to the Coulomb correlation of the electrons, is the dominant contribution to the real part of the surface barrier potential (2.29). A pure image-potential surface barrier can thus be regarded as spin-independent. The evaluation of the phase condition (4.8) for kll = 0 and an image-potential with cut-off (W = 5.05 eV [84]) is shown in fig. 17. The position of the image plane zim = - 0 . 3 9 A relative to the outermost atomic layer is chosen in such a way that crystal- and image-potential induced surface states occur in the energy region where r(.--1 for both spin systems. The intersections of 4B with - 4 c or with - 4 c + 2~r locate the energetic position of the n = 0 or n = 1 surface state, respectively. For the crystal-induced

Z

Fe (110)

/

N

N3 i NI'

N1

OZ.

r

05

c

~

08

1

ix

i

3

-25

i

-2

i

llll/ll~

III

-15

i

J

i

-I

-0

Fig. 16. Bandstructure [38] and energy-dependence of the modulus r c and the phase ~c of the reflection coefficient for Fe(ll0). Dashed lines refer to minority spin, solid lines to majority spin.

0 m < >L3 n" LU Z LU

ILl

u~

A

10 eo

G. Borstel, G. ThSrner / Inverse photoemission from solids"

38

E~

Fe(110)

> LL

45

LU LU

4

0 O3

35 >CO re" LIJ Z LIJ

3 I

25 n--O 2 I

0

II r~

I

I

2~

3~

Fig. 17. Energy-dependence of ~c and q~B in the multiple scattering theory for Fe(110) for an image potential with cut-off (zim = - 0 . 3 9 A). Dashed lines refer to minority spin, solid lines to majority spin. Filled circles indicate the energy of the n = 0,1 surface states, arrows mark the spin-splitting.

surface state a spin-splitting of 0.4 eV is obtained, whereas for the first image-potential state this splitting is only 0.03 eV. The large value for the spin-splitting of the n = 0 surface state can be attributed to the step-like behaviour of the surface potential near the crystal, where the phase q~B varies a s ( E v a c - E ) 1/2. The small spin-splitting of the n = 1 surface state is due to the asymptotic ( E v a c - E ) - 1 / 2 behaviour of q~B near the vacuum level. It should be emphasized that in this example the spin-splitting results only from the different behaviour of q~¢ in the minority and majority spin system, and is thus a consequence of exchange processes in the bulk. Jellium calculations for a spin-polarized semi-infinite electron gas [85] show, that the effective surface barrier potential itself will be spin-dependent due to exchange-processes near the crystal surface. This additional spin-dependence in more realistic barrier potentials will lead to an enhancement of the spin-splitting for surface states. Therefore the simple model discussed above will result in a lower limit for the spin-splitting of surface states on Fe(110). For larger and therefore more realistic values of I Zim ] the n = 0 spin-split Shockley surface state on Fe(ll0) moves below the lower gap edges and becomes a surface resonance. The energy of the n = 1 spin-split image-poten-

G. Borstel, G. Thfrner / Inverse photoemission from solids

39

tial state is simultaneously lowered with respect to E . . . . which in turn results in an increase of the spin-splitting. For a pure image potential with cut-off one then expects spin-splittings of order 0.1 eV (Zim = --1.2 A) or 0.2 eV (Zim = - 1 . 5 A) for the n = 1 surface state at ktt= 0. We note in passing that for kll :~ 0 a klrdependent spin-splitting can occur as a result of spin-dependent effective masses. The structure of the gaps in F e ( l l 0 ) is such that one expects rn~'r > 1, m~'~ < 1, i.e. the spin-splitting of the n = 1 state should increase with ktt. Calculations in the framework of the one-step model of inverse photoemission for Fe(ll0) [34] claimed the existence of a crystal-induced surface state at 2.2 eV in the majority spin system at kit = 0. Since in these calculations the effective surface barrier potential has been approximated by a step potential, results on image-potential induced surface states could not be obtained. The existence of image-potential states on F e ( l l 0 ) near 4.8 eV is known from inverse photoemission experiments [86]. However, these measurements give so far no clear evidence for the existence of crystal-induced surface states on Fe(ll0). An improved experimental resolution, a variation of the detection energy of the outgoing photons and a utilization of polarization effects could clarify this situation. Under these conditions surface states should be clearly separable from bulk states.

5. Summary and conclusions The theory of inverse photoemission from clean metals has reached a level which allows for straightforward application to real systems. Besides giving information on empty electronic bulk states this technique opens up the possibility to study unoccupied electronic surface states and the precise form of the effective surface barrier potential in detail. In this respect future studies on ferromagnetic surfaces should be of considerable interest. At the present level of experimental accuracy in inverse photoemission dynamical self-energy corrections in the effective one-electron potential, screening processes in the optical wavefield near the surface and corrugation in the effective barrier potential seem to be of minor importance, and this fact greatly facilitates the theoretical procedure.

Acknowledgements The authors wish to acknowledge M. Donath, V. Dose, A. Goldmann, F. Hage, F.J. Himpsel and W. Steinmann for discussions and making their experimental results available prior to publication. We thank Mrs. S. Ltithje for her expert job in preparing the manuscript. This work has been financially supported by the "Deutsche Forschungsgemeinschaft".

40

G. Borstel, G. Th&ner / Inverse photoemission from solids

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