J. Phys. C : Solid State Phys., Vol. 12, 1979. Printed in Great Britain. @ 1979
Secondary electron emission from tungsten. Observation of
the electronic structure of the semi-infinite crystal
N Egede Christensent and Roy F Willis$
t Physics Laboratory I, The Technical University of Denmark, DK-2800 Lyngby, Denmark
3 Surface Physics Group. Astronomy Division, European Space Research and Technology
Centre, Noordwijk. Holland
Received 26 June 1978
Abstract. The results of the angle-resolved secondary electron emission (SEE) from tungsten
single crystals, which are described in a previous publication, are analysed in detail in the
present work. The fine structure in the SEE spectra can be interpreted fully in terms of the
electronic structure of the semi-infinite crystal. A large number of elements of structure can
be related to the bulk band structure. In addition, extra features are unambigously interpreted in terms of surface-specific electronic properties. The bulk contributions to the SEE
structure can be interpreted by means of a band model (calculated by the relativistic augmented plane wave method (RAPW)) by assuming conservation of momentum (to within a
reciprocal lattice vector) parallel to the surface during emission. Peaks in the SEE spectra of
bulk origin relate, concerning their spectral position, to high density of states regions in
K-space. All such elements of structure have been identified, and the locations in K space
which contribute have been determined. Some of these peaks have their origin at symmetry
lines, and since this is a consequence of symmetry, and not critically dependent on the
accuracy of the actual band model, it has been possible from the experiment to derive the
energy, K,,as well as K,, i.e. the energy and full wavevector for the emitting states. These
experimental points on the dispersion relation of the electrons in tungsten agree with the
RAPW calculation to within 0.2 eV even up to 25 eV above the Fermi level (EF).
It follows that
many-body self-energy corrections to the bulk bands are small. The calculation covers
energies up to -50 eV above E,, but the SEE experiments cannot give information about the
states above h a , N 24 eV above the vacuum level due to excitation of plasmons and the
consequent smear-out of structure for higher energies. The observation of bulk properties
as structure in the SEE spectra is in fact a consequence of the quantum-mechanical matching
at the surface. In a simple mode, neglecting this matching, the density-of-states contributions
fold out of the expression for the emitted current. The matching conditions, which are discussed in a muflin-tin model, also imply that a number of surface-specific effects are observed. These are referred to as ‘anomalies’. Four ‘types’ of ‘anomalies’ are discussed:
(i) Emission for energies within band gaps even in cases where no surface state exists.
This is explained as emission from vacuum states tailing a short distance into the solid. (ii)
Emission from a surface resonance band (SRB) on the (100) face for energies within the band
gap centred -9 eV above E,. The dispersion relation of the SRB has been derived from the
experiment. (iii) Enhanced emission from lower band gap edges due to a charge enhancement right at the surface. (iv)A particular enhancement of emission from bulk states where the
band structure matches the free-electron dispersion relation of the emitted electrons in slope
(same group velocity). This mechanism assumes that the wavevector lies in a mirror plane,
and is referred to as a ‘vacuum-bulk-resonance-matching’
(VBRM) mechanism. It appears to
be particularly important for interpretation of the spectra for emission from the (111) plane.
The ‘anomalies’ (i) and (ii) represent ‘surface emission’ analogous to that observed earlier
in photoemission experiments.
0022-3719/79/010167 + 41 $01.00 0 1979 The Institute of Physics
167
168
N Egede Christensen and R F Willis
1. Introduction
Although the phenomenon of emission of secondary electrons has been the subject for
a considerable amount of experimental research during several decades, it has only recently been possible to achieve an energy resolution which is sufficient to reveal fine-structure
nzodulation of the otherwise rather featureless background distribution curve it appears
(Burns 1960, Willis et a1 J 971 a, b, 1972, 1974, Willis 1975) that this fine structure can be
related to the one-electron density of states for energies above the vacuum level. Further,
high-resolution measurements using small-angle selective analysers have shown (Willis
et a1 1974) that spectra measured in directions normal to the low-index planes of a tungsten single crystal are closely related to the one-dimensional density-of-states function
along the symmetry lines corresponding to each face. This interpretation is in agreement
with the results obtained from angle-resolved photoemission experiments (Feuerbacher
and Fitton 1970, Christensen and Feuerbacher 1974, Feuerbacher and Christensen
1974).
The experimental work described in Willis et ai (1974)has been substantially extended,
and spectra have been recorded in a number of off-normal directions for each of the three
low-index crystal planes of tungsten. This experimental work is presented elsewhere
(Willis and Christensen 1978) which in the following will be referred to as paper I.
One of the purposes of the present work is to investigate whether the correspondence
between the electronic band structure and the measured secondary electron emission
(SEE) spectra, which the earlier work indicated, is valid in general. It does indeed follow
that the fine structure in the SEE spectra is closely related to the electronic structure of the
metal. It also follows that ‘bulk‘ electronic properties as well as ‘surface’ electronic properties are monitored. Angle-resolved secondary electron emission spectroscopy is therefore an important supplement to photoemission spectroscopy. SEE measures the electronic
states above the vacuum level and the interpretation is not, as in the case ofphotoemission,
complicated by optical transition matrix elements. It should be noted that although
photoemission experiments give information about the states above the vacuum level,
there may be cases where this information differs from that deduced from SEE experiments. In bulk contributions to photoemission (PE), the optical excitation creates a hole
which, when the initial bands are narrow, is localised and therefore may introduce
relaxation effects. This does not occur in SEE spectroscopy. A comparison of the spectral
positions of the states above the vacuum level as derived from SEE and PE for materials
with flat initial bands may therefore give direct experimental information about these
‘relaxation shifts’.
As a detailed analysis of the experimental SEE spectra in terms of a band model was
necessary, we have extended an earlier relativistic augmented plane wave (RAPW)
calculation (Christensen and Feuerbacher 1974) to higher energies (up to 50 eV above
the Fermi level). This band calculation is used to derive bulk SEE contributions in a model
which assumes that the current is produced by Bloch waves propagating through the
surface from the bulk such that the momentum parallel to the surface is conserved during
emission. It includes ‘Normal’ as well as all relevant ‘Umklapp’processes, where ‘Normal’
refers to states with wavevectors in the first Brillouin zone and ‘Umklapp’ processes
include reciprocal lattice vectors different from 0. The calculations yield information
about spectral positions of various elements of structure as well as their origin in K
space. The locations in K space are defined by the surface momentum conservation
condition mentioned above together with the requirement of energy conservation. These
Secondary electron emission from tungsten
169
spectra are essentially density-of-statesfunctions related to restricted regions in K space.
They will be referred to as densit4’-of-contributing-stu~es( D C S ) functions.
Apart from the elements of structure that can be related to the band structure through
the DCS functions, a number of additional features are observed. These ‘extra’elements of
structure are referred to as ‘anomalies’,a term which is solely used to distinguish them
from DCS structure. The ‘anomalies’ include emission from: (i) evanescent states, i.e.
states for which the vacuum wavefunctions tail into the solid; (ii) emission from surface
resonance bands; (iii) enhanced emission from states at lower band gap edges (see paper
I; and (iv) enhanced transmission through the surface in cases where the band structure
matches the dispersion relation for the emitted free electrons perfectly, i.e. the dispersion
relation has the same slope in the ‘initial state’ in the solid as in the ‘final state’ in the
vacuum region. The observation of emission from surface evanescent states in the SEEspectra supports interpretations of earlier photoemission results (Feuerbacher and
Fitton 1972, Christensen and Feuerbacher 1974, Feuerbacher and Christensen 1974)
This type of emission is sometimes referred to as ‘surface emission’, a term which should
not be confused with surface optical effects in photoemission. The emission (i) appears
within as well as outside band gaps, but in the band gap regime it represents the only
emission mechanism (disregarding surface resonance bands and emission following
thermal diffusescattering). It is therefore also sometimes (Feibelman and Eastman 1974)
called ‘band gap emission’.
The present paper is divided into five main sections. In 52, a simple model for the
emission process is presented. It contains a discussion of the wave-matching procedure
at the surface in cases where the crystal potential can be reasonably well described in a
muffin-tin model. Section 3 is subdivided into two parts. Section 3.1 describes the approach which we take to analyse ‘bulk contributions’ to structure in the SEE spectra,
whereas 53.2 describes the so-called ‘anomalies’. The detailed analysis of the spectra
follows in $4,which has five parts. The first part, $4.1, describes geometrical features of
the analysis of off-normal spectra. The analysis of spectra from the three low-index
faces (loo),(110), and (111) follows in w . 2 , 4.3 and 4.4. Section 4.5 contains a crosscheck of the results presented in w.2-4.4 and experimental dispersion relations for
electrons in tungsten are shown and compared with theory. Summary and conclusions
are given in 5.5, where the relevance of the present results to photoemission experiments will also be briefly discussed.
2. Secondary electron emission through a surface
Although a major result of the present work is the demonstration of a close relation between structure observed in the SEE spectra and the bulk band structure, it should be
emphasised that the SEE process of course is a surface effect. The electrons observed at the
detector have indeed passed the surface of the crystal, and the surface necessarily influences the current. Quantum-mechanical matching conditions, which must be fulfilled at
the surface, imply that the surface acts as a ‘transmission filter between the bulk and the
vacuum region outside the sample. In addition, special surface electronic properties
(surface resonances etc) will manifest themselves in the observed spectra.
The present section will be devoted to a discussion of a simple model of the emission
process. Although simple in principle, the practical aspects are so involved that we did
not use it for the actual calculations. The presentation serves the sole purpose of illus-
170
N Egede Christensen and R F Willis
trating to some extent what physical properties are seen in SEE spectra and to suggest a
possible way of extending future calculations.
A model for the SEE process may be constructed along lines somewhat similar to those
applied by Nicolaou and Modinos (1974) and Modinos and Nicolaou (1976)for the field
emission problem. However, we do not need to apply a transfer-Hamiltonian procedure.
We assume that the primary electron beam and a number of decay mechanisms have
created a ‘continuum’ distribution of ‘hot electrons’ in states above the Fermi level in
the semi-infinite metal. These are described in paper I. The SEE current is obtained by
calculating the usual quantum-mechanical current through the surface using a reasonable representation of the eigenfunctions for the Hamilton operator of the semi-infinite
crystal. These wavefunctions are obtained in the following way:
The crystal is visualised as filling the half-space x > 0 and terminated abruptly at
x = 0 (the surface). The potential is assumed to have a step at x = 0. The region x < 0
is the vacuum region. The surface, which is assumed to be parallel to atomic layers, is
located i d outside the outermost atomic layer, 6 being the distance between successive
layers.
The crystal potential for x > 0 is described by the muffin-tin model. This model is
also used in the actual band calculation, and it enables a rather simple formulation of the
wave-matching procedure at the surface. This is because the wavefunction in the interstitial region is plane-wave like. To use this property we assume that the surface runs
through interstitial regions only, i.e. regions where the potential is constant (the mufintin zero, V,).
Choosing the zero of the energy scale at the vacuum level, and denoting by E ,
and & the Fermi level and workfunction respectively, the Hamiltonian for the semiinfinite crystal is
where V, = -(EF + (ps). The vectors R,, are the lattice vectors and the sum gives the
muffin-tin potential (c(d) = 0 for d = ( Y - Rn/ > Rs where Rs is the muffin-tin sphere
radius). In this Hamiltonian we neglect surface potential effects (mirror potential etc).
These could be included in a refined model.
In order to determine the number of electrons that traverse the surface (x = 0)
per unit time and unit area we must calculate
(YVY* - Y*VY).dS
where the wavefunction Y satisfies
HY = EY.
(3)
(The summation indicates that we must sum over all states { M ) with energy E.) The
integration (2) is over the plane x = 0.
The wavefunction inside the metal will be assumed to be of the same type as used by
Nicolaou and Modinos (1974) and Modinos and Nicolaou (1.976) namely a sum of a
Secondary electron emission from tungsten
171
wave Y B incident
I
on the surface from x = cc and a number of reflected waves Y j :
Y = v-l’2YBI +
v-112
1CjY
x > 0.
(4)
j
The wavefunction YBIis a Bloch wavefunction of the infinite crystal corresponding to
the energy E (the eigenvalue considered in equation (3)). The summation includes waves
of the complex band-structure problem for the infinite crystal, but the sum should only
include those states which propagate or decay (evanescent waves) in the positive x
direction. Vin (4) is a normalisation volume.
The wavefunctions Y j are, in the interstitial regions, superpositions of plane waves :
Yj=
C {U!:
9
exp[iQg+ ,( r - Ai)]+ U&exp [iQ;(r - A , ) ] }
(5)
where we refer to the region between layer i and (i - 1).The coefficients U satisfy
.
U!:+
= U,: exp(ikj a , )
(6)
and a1 is an interplanar primitive lattice vector. The vector A i in (5) is ( i - l)al. The
surface region corresponds to i = 1. The vector k i l = (0, k,, k,) is the wavevector parallel
to the surface, and it is restricted to the first Brillouin zone of the 2D surface lattice.
The vectors g = (0,g,, 9,) are reciprocal lattice vectors of the 2D surface lattice. The
wavevectors Q are defined as
QB
=
(+[(2m/fi2)(E- Vo)- (kli
+ g)2]1‘2,kll + 91.
(8)
The wave incident on the surface from the interior of the metal is of similar form:
(U:: exp[iQr
YB,=
.(U - Ai)]+ U:;exp[i
Qb- .( r - Ai)]}.
(9)
9
It should be noted that, for a certain k there may be several k\ values, i.e. several incident
waves corresponding to the same energy E. The normal components, k,, of the wavevectors may, for the Y j functions. be either complex or real. A real k , corresponds to a
propagating wave, whereas a wave with complex k , is evanescent. The incident waves,
which are coming from x = CO, are always propagating, i.e. they have real k , components.
For energies in a genuine gap in the band structure there will be no incident waves YBr
The totai wavefunction is constituted by ‘reflected’waves CCjsYjdecaying into the metal.
They may, under circumstances which we can formulate in a simple mathematical
condition later, form surface state or a ‘surface resonance band’. It is of conceptual importance to realise that the summation in (4) in general, i.e. for energies within as well
as outside a band gap region, can contain states with complex k,, i.e. evanescent waves,
waves ‘tailing into’ the solid.
The wavefunction Y in the region x < 0, i.e. outside the metal may be written as
Y = V-’’’
1Bg exp(iqgx)exp[i(kl, + 9 ) . p]
x<o
(10)
+ g)2]112.
(11)
@
where
p = (0,y, z )
and
q, = [(2m/h2)E- (kil
So far, three sets of expansion coefficients have been introduced, the U’s the Cis
and the B, coefficients. The coefficients U are in principle determined by solution of the
(complex) band-structure problem. The other coefficients (Cj and Bg) are obtained by
172
N Egede Christensen and R F Willis
matching the wavefunctions in (IO) and (4) at the surface (x = 0). This matching condition
requires continuity of Y and its derivative with respect to x at x = 0. This matching
implies that (layer index i = 1)
B,
=
U::
+ U!; + 1 Cj(U{', + U{,)
.i
and
If n g-vectors are included, then (12) represents a set of 2n equations with 2n unknowns,
n C j coefficients and n B, coefficients.
In order to evaluate the current we use (2),but we must, for a particular energy E,
sum over all states that can contribute, i.e. over all k\ and all k l l .The summation of
k l l will be an integration over the 2D Brillouin zone, or rather over that part which is
relevant when the detector position and detector solid angle are specified.
If we want to sum a function F over all metal states { M ) , this is done by the simple
transformation :
Therefore, using the wavefunction (IO) we get from (13) and (2)
where
'Ij-11 - (kll + s)l= S[P, - (ky + S , ) l ~ [ P Z- (kz + 9,)l
e = sin-' ( ~ ; i h ~ / 2 m E ) ~ " , 4 = tan-
P 2 = 2mE/h2,
(Pz/P,)3
P
=
( P x , P p P,),
and
P 11 = (0,P,. P A .
This is the current in the direction specified by the polar angle e (measured from the
surface normal) and azimuthal angle 4 of electrons with energy Eo. The vector p l ,
is the parallel component of the momentum of the emitted electron. In deriving this
result, we have assumed that E is not in a band gap, i.e. that we have propagating incident
wave solutions $, with normal k components ki. The expression for the current when
Eo falls in a gap regime has the same structure as (14) if the summation &is omitted and
the denominator replaced by 1 (the constant factor in front is also different).
A full-angle measurement would detect, at the energy EO, a current j (EO) given by
Secondary electron emission from tungsten
173
In this expression
=
0
{1
5<0
< > 0.
It follows from (14) and (15) that neither the directional measurement nor the fullangle experiment seem to measure the full bulk density of states function or the surface
density. The surface acts as a filter, transmitting only certain components of the wavefunctions, and the transmitted amplitudes are further weighted by the reciprocal normal
component of the band state group velocity and the square-root factor. A transitionmetal wavefunction requires inclusion of many g .vectors and as only few g components
will be transmitted, we can see that, even disregarding the square-root factor in (I 4) and
(15),there would appear to be little reason why the bulk (or surface)density of states should
be reflected in the final current.
To illustrate and explore this more clearly, let us calculate the local density at the
surface. This is done by setting
F
=
I$(x
=
0, J , z)I2 6(E - E M )
(16)
in (13). The local density p(E, x = 0, y, z) is then
The average density at the surface is
A is the area over which the integration is performed, and we get
Here
By comparing (15), (19), and (20)t we see that in those cases where the energy Eo is
sufficiently large compared with the square of the longest g vector of importance in
expansion of the wavefunctions, the full-angle spectra are essentially measuring the
average local density at the surface; the modulation by the square root is rather smooth.
Structure in the current versus EO curve relates to structure in this average density of
states. But this relationship fails if 'relatively long' g vectors are required. In the case of
angle-resolved measurements (14), even fewer g components can pass the 'surface filter'
and the current will have even less similarity with the density of states functions.
It should be realised that the results derived above are conceptually rather simple,
but an actual numerical calculation is quite involved. Further, the calculations leading
to (14) and (15) depended essentially on the application of a crystal potential of the muffintin type. It is well-known that the muffin-tin model works quite well in connection with
bulk band structure calculations for close-packed structures, especially for FCC lattices.
t The expression (19) reduces to the usual bulk density-of-states function if the wavefunction $ is replaced
by a bulk Bloch wave.
174
N Egede Christensen and R F Willis
The BCC structure (which is the structure of tungsten) is not as closely packed as the FCC
lattice, but still band calculations only require moderate corrections for non-muffin-tin
terms. The inaccuracies due to the assumed form of the potential in a calculation of (14)
and (15 ) will however be considerably more serious than those met in conventional bulk
band calculations. The entire discussion assumed that the crystal could be separated by
planes running midway between atomic layers parallel to the surface, and that these
planes are entirely in the constant-potential region. Therefore, the present model requires the muffin-tin spheres to be considerably smaller than those used in the bulk band
calculations.
In usual band calculations we can choose the mufin-tin radius in the BCC lattice to be
RSmax= aJ3/4 spheres touching along (1 11)). In the model discussed above R, d a/4
in the case of (100) and (110) planes and R, < aJ3/24 when the (1 11) plane is considered. It may be reasonable to use the model for emission from (100) and (110) planes
(the field-emission calculations by Nicolaou and Modinos (1974) and Modinos and
Nicolaou (1976), indicate this) but its application to the (111) case is quite dubious. We
expect that even the qualitative features which can be derived from (14) and (15) do not
apply to (111) emission. We see that wave matching at the (111) plane becomes much
more complicated than in the other two cases and serious discrepancies between the
present simple theory and experiments are to be expected in this case.
Briefly summarised, the results of this section show that a full-angle SEE experiment
measures a current (equation (15)) which, when the energy is sufficiently large, is closely
related to the average local density at the surface (equation (19)). In this context, by
‘sufficiently large energies’ we mean kinetic energies which are so large that all g vectors
which are essential in the construction of the wavefunctions are transmitted through the
surface. It follows, for example from equation (19), that the average density at the
surface contains hulk as well as surface Specific properties. In SEE experiments which
are angle-resolved, further restrictions are imposed on the number of g components
which appear in the detected current. Therefore, it must be anticipated that the angleresolved SEE spectrum, even at high energies, differs more in spectral shape from the
average density function (19): i.e. there appears to be a more eficient ‘surface filtering
effect’. Only detailed numerical calculations, for example using the method outlined
here, can show how effective this ‘filtering’ is.
It does follow, however, that in the angle-resolved case we may still expect to see
density-of-states related features, such as peaks originating in the bulk band structure
where BElak, = 0. This means that structure due to band edges in the K space curves
(discussed in 4 3) is expected to appear in the SEE spectra. Lineshapes and actual amplltudes are of course determined by the ‘filtering’ and the values of the B, coefficients as
determined by the wave matching. The angle-resolved SEE spectra will contain, as the
full-angle spectra, surface- as well as bulk-specific contributions. As follows from paper I,
and the discussion in the following sections, much structure in the SEE spectra can be
interpreted in terms of bulk density-of-states type functions. Although these features
relate to bulk properties, it should be emphasised that their observation is a consequence
of the wave matching at the surface. Their occurrence cannot be fully understood in
terms of models that consider only bulk transport properties and neglect the specific
transmission properties of the surface.
3. Origin of structure in
SEE
spectra
Paper I contains a broad phenomenological discussion of mechanisms which introduce
Secondary electron emission from tungsten
175
structure in the SEE spectra. We shall give here a more detailed presentation of some of
these features, and in particular present the lines that will be followed in the analysis
in terms of the band structure.
For the reasons outlined in paper I and, at this stage, in spite of the reservations noted
in $2, it will be assumed that part of the elements of structure can be related to structure
in appropriate density-of-states functions. An attempt is made to relate the observed
peaks to regions in k space with high densities of bulk states. This analysis concerns only
spectral positions of structure, not lineshapes nor intensities. Elements of structure which
cannot be interpreted in this way will be referred to as ‘anomalies’. This classification is
somewhat misleading since the ‘anomalies’ represent physical effects of general character.
This section is divided into two parts. Section 3.1 describes the principles of the method
used in the analysis of bulk contributions to the structure in SEE spectra, whereas the
‘anomalies’ are discussed in 0 3.2.
3.1. Approach to analysis of bulk contributions
In $ 2 it was natural to consider states in the two-dimensional reciprocal space. Now we
want to discuss bulk properties and apply the bulk band structure. It is thus most convenient to change to the usual three-dimensional k space. This means that the symbol k
when we consider bulk properties means a wavevector in the first Brillouin zone of the
three-dimensional lattice. Reciprocal lattice vectors of the 3D lattice are called G , and
we use K to denote a vector of the type k + G, where G # 0. It should be noted that
we can always relate states in this 3D momentum space to states in 2 D space if we also,
in the latter, specify k,. This is true since the 2D reciprocal lattice is a projection of the
3D reciprocal lattice on the plane considered. This means that the 2D Brillouin zones
can be placed within the 3D Brillouin zone (BZ) in such a way that the borderlines of the
2D zones are lying in the faces of the 3D zones. The reduced two-dimenional zones have
been constructed for the three low-index planes of the BCC lattice, and they are shown in
figure 9 of paper I together with the usual BZ of the 3D lattice.
The bulk density-of-states contributions to the fine structure in the SEE spectra are
calculated from the RAPW band structure in a model that assumes that the component of
the momentum parallel to the surface is conserved during emission. Thus, if the detector
collects electrons with momentum p , it is assumed that pll = hKlIwith K l l = k l l + GII,
where Gll is the parallel component of any reciprocal lattice vector G, and k , , is the
parallel component of the reduced wavevector k for the electron in the crystal. In the
vacuum region an emitted electron has the kinetic energy Ekin= p2/2m, and since we
assume that energy is conserved during emission, this implies that the energy level in
the crystal of the same electron is E , = E , + 4s + Ekln,& being the workfunction.
The basic assumptions are then
The parallel momentum pllis
p l , = ~in0(2mE,,~)”~.
(22)
The angle 0 is the detector angle measured from the surface normal. The function
E,(K) is the dispersion relation as given by the RAPW calculation. Each state E,(K)
176
N Egedc Christensen and R F Willis
satisfying (21) and (22) contributes to the DCS function by an amount which is essentially
l/lVK(Ef(K))l;i.e. they will give a contribution that is proportional to the inverse group
speed in the state K .
The calculations are therefore performed in the following way:
(i) The detector position relative to the emitting surface is specified. This is done by
choosing the two external angles 8 and 4. The angle 8 is a polar angle measured from the
surface normal. The azimuthal angle q5 is measured from a reference line which lies
within the surface plane considered.
(ii) The kinetic energy Ekinis varied in steps of 0.067 eV from 0 to 40 eV. The band
energy E, = E , + q5s + Ekinand external momentum p corresponding to each value
of Ekinare determined.
(iii) Knowing pI1we then search k space for points where k(k + G),l = p i ,and where
the band energy E ( K ) = E,.
This is done by measns of a root-finding procedure which solves the equation
with respect to K,, the wavenumber perpendicular to the surface for each band, n, in the
relevant energy range. The band energies appearing in equations (23) are obtained, by
means of the three-dimensional interpolation procedure?, from the RAPW band structure. The wavevector search is performed over a K space region corresponding to seven
shells of reciprocal lattice vectors G. The contributions originating in the zone with
G = 0 are referred to as 'normal processes' (N-'processes)and the others as 'Umklappprocesses' (U-processes).
(iv) For each wavevector K = K,(n, Ekin,p , , )= KnlL + K O , / ;for which Krill is root
number I of equation (23) in band n, we then determine the group velocity
The
contribution to the spectrum from each point is then given by
ANn,
l(Ekin'
2'
4) = w/ l y ,I /
(24)
where the weight factor W is 48 divided by the order of the group of the wavevector, the
calculations being performed in the 1/48 irreducible zone. Finally, the 'total' amplitude
N(Ekin,1!3,+)is obtained by summing:
N(Ekin,
' 5
4) =
l(Ekin'
7
'
4)'
(25)
n, 1
The calculation gives not only the spectrum (25), the k space points, the band numbers,
and the relative weight corresponding to each contribution are also identified.
t The RAPW band energies are evaluated at 285 k points uniformly distributed over the 1/48 of the irreducible
Brillouin zone. In order to be sure that all roots of equation (23) are found it is necessary to scan the K, range
using a rather fine mesh, and for evaluation of the left-hand side of equation (23) it is necessary, by means of
symmetry operations and lattice translations, to reduce to the irreducible zone before the three-dimensional
interpolation procedure is applied. The fact that we use this interpolation method implies that the computing
time necessary for calculating a spectrum covering 600 values of the kinetic energy and including seven shells
of reciprocal lattice vectors becomes very long. Large-angle spectra required approximately 40 min CPU
time on an IBM 370/165 each, whereas the time required to obtain (100) and (110) spectra for small 0 was
considerably less. However, the amount of information produced by the calculations is quite extensive, and in
addition to the results described here only small modifications of the SEE programs were necessary to compute
angle-resolved bulk photoemission spectra for a large number of photon energies simultaneously. This was
done almost without additional expense of computing time. These photoemission results will be discussed
elsewhere.
177
Secondary electron emission from tungsten
Figure 1 is a sketch demonstrating how the resulting K space plots in the following
can be used. The curve N ( E ) represents the SEE spectrum, or a part of it, and the parabola
pI,(E)gives the parallel component of the momentum p of electrons emitted with
kinetic energy E. The left-hand part gives for each band (numbers n and m) the curves in
reciprocal space from which the contributions to the spectrum originate. The K space
plot is placed so that the ordinate axis is giving the parallel component of K . As an
example of use of the diagram, the peak A in the SEE spectrum may be considered.
The kinetic energy is E,, and the parallel momentum is pIIKBy drawing a horizontal line
through the left-hand part of the figure, it is seen that band number n contributes to
peak A at Knl, and band number m contributes at K,, and K,,.
For illustration it may be worthwhile to show how the K space plot and IXS function
will look for a free-electron model. The bands in this case are given by E,(k) = hk2/2m,
'N
Kinetic energy E
Figure 1. Sketch demonstrating the use of 'K space plots' in connection with analysis of
off-normal spectra. The conservation of parallel momentum (equation (21))ensures that a slice
in K space is scanned. The plane of this slice is spanned by the KIIand K , directions. The
parabola shown in the right-hand diagram represents the parallel momentum of the emitted
electrons as a function of kinetic energy kil= sin 0 4 2 m E ) . 0 being the emission angle).
The peak A at the kinetic energy E, corresponds to electrons having the parallel momentum P , ~ , , and thus the parallel wavevector K l l A
= pllA/fi. The left-hand diagram shows
K space root curves' (solutions of equation (23)) for two bands n and m. The horizontal
line pi, = pllAintersects these curves at Knl, Kml,and KmT These points are then the locations in K space contributing to the structure A. The theoretical curve N ( E ) is obtained from
equation (25), and would in the example shown here contain properly weighted density-nfstates contributions from the three points K,,, Kml, and Km2.
E , being measured from the band bottom. The kinetic energy is
If the emission angle is 8, measured from the surface normal, then
hKll = pll = ~in8(2mE,,,)'/~
N Egede Christensen and R F Willis
178
which together with (26) gives the K space curve through
K f - (K,,/tanO)2= 2mEv/h2.
(28)
Thus, in the free-electron case, the,K space curve related to a certain emission angle O
is a hyperbola with K l i = tand K,. as an asymptote.
This is shown in figure 2, where the density-of-contributing-states (DCS) function is
also shown. The DCS is
KII
N
Free-elect ron model
t
Kinetic energy ( € 1
Kl“
Figure 2. K space plot for a free-electron model, where the vacuum level is E,, measured from
the band bottom, The root curve is a hyperbola (equation (28)) and the asymptote forms the
angle 0 with the K , axis. This angle is the (external) emission angle 0. (In the case where
E , = 0, the hyperbola will degenerate to this straight line.)
3.2. ‘Anomalies’ in SEE spectra
As mentioned earlier, we refer to structure in the SEE spectra which cannot be related to
high density-of-states regions in the bulk band structure as ‘anomalies’. These effects
may include
(i) Emission from evanescent waves, i.e. waves ‘tailing’ into the solid.
(iij Emission from states within ‘surface resonance bands’.
(iii) A particular enhanced emission from lower edges of band gaps.
(iv) Enhanced emission due to a ‘vacuum-bulk resonance’.
The contribution (i) will be assumed to contribute to the rather uniform background,
and does not introduce particularly strong elements of structure. This type of emission
will contribute inside as well as outside band-gap regimes. For energies within a band
gap the emission will be entirely due to the effects (i) and (ii). Even if thermal diffuse
scattering can be neglected, this means that the intensity will not go to zero in the band
gap region. If a surface resonance band exists there may even be structure in the SEE
spectra for energies within a band gap (Willis et a1 1977). ‘Surface emission’ is a term
sometimes used for effects of type (i) and it has also been observed in photoemission.
The existence of a surface resonance band can be discussed within the model described
in 0 2. For energies within a band gap there are no incoming bulk Bloch waves $Bl, and
Secondary electron emission from tungsten
179
the wavefunction is ( A being a normalisation area)
*=
{
CjI+bj
x>o
j
A-'/'
1
B, exp(iqgx)exp[i(kil + g ) . p ]
B
x<Q
(30)
where k i iand g have the meanings of 0 2. If we define the coefficients FL and DL through
There are n reciprocal lattice vectors g. The g vector number m is called gm,and
further we introduce the abbreviation clgm as
= [(2m/h2)(E - V , ) - (kl,+ gm)2]'i2.
a,,
(33)
The conditions (32)then imply
C,Dil
+ C2Di, + ... + C,Di,
=
B,,
+ . . . + CnDim= B,,
n equations
(344
C,Di, -I-C2Dim
and
C,Fh,
+ C2Di, + . . . + C,Fi,
= (4,1/~gl)B,l
C&,
+ CJi, + . + C,Fi,
= (qgm/agm)Bgm
n equations.
* *
(34b)
A set of n homogeneous equations for the C coefficients is obtained from (344 and (34b):
If the coefficient matrix
{'rs}
E
{Der - ( a q J q g r F i r 1
(36)
has a vanishing determinant
det{AJ = 0
(37)
180
N Egede Christensen and R F Willis
then (35) has a solution C,, . . . Cn. This (37) is then the condition within the model of
$2 that there exists a surface state or a surface resonance band. The dispersion relation
is obtained from equation (37).
The third 'anomaly' (iii) is due to a 'pile-up' or charge at the surface in states at the
lower band gap edges ('bonding states'). This effect is discussed in paper I.
To explain the last 'anomaly' (ivj we consider the one-dimensional problem of normal
emission from a semi-infinite crystal. The model of $ 2 is again considered (see also
Christensen and Willis 1978). There we had chosen a certain normalisation volume V
for the wavefunction for x > 0 (the metal region). Arbitrarily we use the same normalisation volume for the wavefunction in tlie region x < 0. Another choice would merely
rescale the B coefficients. It is important, however, to notice that once normalisation
volumes have been chosen and IC/ is normalised for x < 0, then ) I is normalised for
x > 0 as well due to the matching conditions. The wavefunctions cannot be scaled
arbitrarily with respect to each other but are linked together by the matching procedure.
For a free-electron wavefunction as well as for a Bloch wave IC/, we have
where Vis the normalisation volume for $r As used in (2j, the current through a plane at
x = xo parallel to the surface is
Letting A be the area of the normalisation cube (volume Vj parallel to the surface, we
have? for x < 0
aE
dp,
--
---j($*;$
1 h
A2mi
a
- $%$*)
dydz
where p , = hk,(x < 0) and the integral is evaluated over a plane of area A but with the
plane being placed at any xo ( < 0). Now, it may happen that for some particular energies
E we find that the 'incident bulk states' are such that
h(aE/?p,)= dE/alz:,
(41)
i.e. the derivative of the bulk band structure is the same as that of the free-electron parabola giving the dispersion relation for the emitted electrons. With the normalisation
t This is easily seen by applying the time-dependent Schrodinger equation and Green's theorem:
which implies that
(Green's theorem). The surface S is the boundary of the volume K
Secondary electron emission ,from tungsten
181
chosen, this means that the incident wave carries the same current as the vacuum state.
Thus, in this case I/' = I! and the reflected waves carry no current. Phrased otherwise,
the entire current towards the surface carried by the incident propagating bulk state is
transmitted through the surface, i.e. the transmission probability reaches a maximum.
This situation is illustrated in figure 3(a).The bulk bands are plotted here in a nearlyfree-electron (NFE) model. This is the simplest case for which the above mentioned
'~acuum-bulk-resonance-matching' (VBRM) effect can be observed. The free-electron
NFE
model
lo1
Figure 3(a)Nearly-free-electron (NFE) model (heavy lines). The band gap is Ea at the zone face
G / 2 ) . Parabolic dispersion relations having their minima at the vacuum level have been
placed such that the, match the bulk structure at the p o i n t s c a n d 3 : . At these points
the so-called vacuum-bulk-resonant-matching (VBRM) condition is fulfilled. The \FE model
represents the simplest band model where this condition can be fufilled. The situation shown
here corresponds to normal emission from a plane having G as a normal direction.
N
Figure 3ib) K space plot in the NFE model corresponding to emission in the direction U mea-
sured from the surface normal (G).The plane spanned by K , , and K , is assumed to be a mirror
plane, The gap (Eg)extends from thc lower edge (A) to the upper edge (B). The p o i n t s a a n d
@are points where the VBRM condition is fulfilled. The curve N ( E ) contains the gap-edge
peaks A and B. and in addition the VBRM peaks C and D.
182
N Egede Christensen and R F Willis
parabola has its zero in energy at the vacuum level, and in the figure the parabola is
below the lower
drawn so that the matching (VBRM) condition is fulfilled at a point
band gap edge. By displacing it further, it is seen that above the upper edge there is
also a possibility of such a matchingo. Note that the normal component of the wavevector is not conserved. This means that the free-electron parabola can be placed with
its minimum at E" anywhere on the K,-line. Further, it is obvious from figure 3(a) that
the VBRM condition cannot be obtained if the band structure itself is purely free-electron
like. The matching condition requires bands that are distorted, i.e. a finite crystal
potential is necessary.
In cases where the emission angle 8 is different from 0 (off-normal emission), the points
in K space fulfilling the VBRM condition are also easily identified. In figure 3(h) is shown
a K space plot for a NFE model. The VBRM points are determined as the points where
the K space root curves (as described in fs 3.1) have a tangent that forms an angle 0 (the
emission angle) with the K , axis. The constant energy sphere corresponding to the energy
Ginof the emitted electrons is centred on the K , axis at K, = KYo which is the 'nonconserved' part of the normal wavevector; i.e. the crystal surface must supply a normal
momentum Ap, = -hKyo. Figure 3(b) also shows qualitatively how the VBRM may
modify the spectrum by introducing peaks below and above the band gap edges. In more
complicated band models there may appear VBRM structure far from band edges. This is
the case for tungsten as will be demonstrated later.
To conclude this section it is stressed that the VBRM is not a result of the bulk band states
being free-electron like. Thus it is not a result of a particularly large plane-wave component corresponding to a certain reciprocal lattice vector. In fact VBRM is impossible in a
free-electron model due to the finite value of the vacuum-level energy E, when compared
with the band bottom in the empty lattice. Further, it follows from the discussion above
that the VBRM condition can be derived only by imposing wave-matching conditions at
the surface and thus provides the linking between vacuum wavefunctions and the wavefunctions inside the metal. It is in this context immaterial whether we use a mufin-tin
model or not. The VBRM is therefore not dependent on the actual details of the model of
$ 2 and in particular it follows that the arguments are equally valid for all crystal faces,
even the (111) face where the muffin-tin model was expected to be rather inadequate.
0
4. Analysis
4.1 .'Geometry'
This section describes the bulk band structure contributions to the fine structure modulation of the SEE spectra. The calculated spectra are essentially the state densities evaluated at points in K space where hK,, = p , , ,(i.e. the parallel momentum is conserved)
and K = k + G, G being a reciprocal lattice vector. The normal component K , is
obtained from the band structure as described in $2. The detector position is specified
by the external angles 6 and 4. The polar angle 8 is measured from the surface normal,
whereas the azimuthal angle 4 is measured from a reference direction in the plane considered. This reference is for the (100) plane as well as for the (110) plane, the 1001/
direction. For emission from the (111) plane, the reference (4 = 0) direction is (1 12).
The experimental spectra are all taken for 4 = 0", and 8 = o", 10", 20", 30°, 40°, 50",
60", and 70", i.e. there are eight experimental spectra for each face. The calculations have
been carried out in all these cases and in addition for non-zero azimuthal angles. The
calculations for 4 # 0 will not be discussed extensively here. Below q5 = 0 unless we
explicitly specify, 4 # 0.
183
Secondary electron emission from tungsten
The calculations were performed in a ‘repeated’ zone scheme including seven shells
of reciprocal lattice vectors G. The contributions with G = 0 are referred to as ‘Normal’
process contributions (N) and those with G # 0 as ‘Umklapp’ (U) contributions. The
seven shells of G vectors contain a total of 87 reciprocal lattice vectors. This does not
mean, however, that the spectra will contain 86 different U-contributions. The number
of possible U-processes will depend on the angle 8 and the energy. For sufficiently low
energies and small angles 8, only N-processes can contribute to the (100) and (110)
spectra; all possible U-processes will just be replicas (apart from intensity) of these spectra.
As will be shown below, even up to EkinN 40 eV and 8 70” only very few different Uprocess spectra need to be considered. The (111) spectra will always, i.e. for all 8 and
Ekin,include U-contributions, but in the energy range which we consider, the number
of different U-spectra will be very small.
Since the parallel momentum is conserved during emission, only a ‘slice’ in K space
is scanned when 6 is varied. In the (100) case the ‘slice’ is the K$, planet (a plane
spanned by two TH directions). In the (110) case the plane which we scan is spanned by
a TN and a TH direction and the same plane is scanned in the case of (111) emission.
(In the surface Brillouin zones we are scanning A, C, and T, lines.) The slices related to
the (100)and (110) spectra in K space are shown in figure 4. Figure 4(a) refers to the case
-=
I-
1
0
- -
- 8
I
IO
20
30
LO
Kinetic energy,EkV)
(C)
Figure4. Sections in K space scanned, with the present choice of azimuthal angle, in the
cases of (100) emission ( U ) and (110) emission (b).(c) shows pi, as a function of kinetic energy
for the emission angles 0 used in the experiment. These figures serve, as discussed in the text,
the purpose of identifying the various Umklapp processes which are possible in certain energy
ranges and for certain choices of emission angle.
of emission from the (100) plane. As long as K l , is less than 2n/a, where a is the lattice
constant, only two types of spectra need to be calculated, namely the ‘N’ type and U(101).
All other zones in the (1,0,0) direction will just give replicas of these spectra. From the
symmetry (figure 4a) it is also obvious that only 0 d K , < 2n/a needs to be considered.
When K l ,is increased beyond 2n/a it is seen that U(002) must be taken into account. If
Kll < 4n/a, U(002) is the only new spectrum which we need to consider in addition to
‘ N and U(101), since U(202) etc will be replicas of U(002). The range of kinetic energies
covered by the calculation is 0 < Ekin< 40 eV and for all angles 6 5 37” we then have
t Now the x, y , and z directions refer to the (10@), (010) and (001) crystallographic directions.
184
N Egedc Christensen and R F Willis
K l l < 2n/a in the entire energy range. Thus for these angles we have only two different
types of spectra to consider, 'Normal' and U(101). For angles 8 > 37" U(002) will contribute at the high-energy end of the Ekinrange, but for Ekin< 40 eV we will never reach
K l l= 4n/a Figure 4(b) shows the K-space slice scanned in the (110) case. For combinations of 0 and Ekinsuch that K I I< n/a (NP) there will be only one type of contribution
to the bulk spectra, the "'-process spectrum. The U(110) (and all others in the (1, 1,O)
direction) will just be replicas of the N-contribution. In fact, only 0 < K , 6 5(2)n/a
(TN) need to be considered, the N P P line is a mirror line. U-processes become important
when K I Iexceeds n/a where U(1,0, 1)-contributions start. This is the only U-contribution as long as n/a < K i l < 2n/u. When K l l exceeds H (2n/a), U(002) appears. For
Ekin6 40 eV the only U-processes that are important are U(101) and U(002), the latter
even only for 0 > 37". When 0 < 15" only N-processes are contributing to the bulk
(1.10) spectra in the entire range 0 d Ekin6 40 eV. Figure 4(c) shows p l ,as a function of
kinetic energy for all experimental angles 8. This figure, together with 4(a) and 4(b)
then allows identification of the U-processes important in any energy range for
each angle 8. We do not pay much attention to the experimental spectra for energies
above Emex= h o p 1: 24 eV (see paper I).
A similar analysis for emission from the (111) crystal plane can be made from figure
5. As mentioned earlier, the calculations include seven shells of G vectors. This means
that G = (2,2,2) 2n/a is the largest G vector included, and therefore U(321) (see figure 5)
is not included. Thus, in the (111) case, our calculations only exhaust all possible Uprocesses as long as ti is less than Kll, = $ 5 J6n/a = 1.63n/a, and the calculated
H
U12201
- *
r
-
-
N
I
- ----
(1101
- 4
I
Figure 5. K space section scanned by emission from the (111) surface. The plane of this
slice is (as the planes of the slices of figure 4) a mirror plane. Note, however, that the ( J 1 1)
plane is not a mirror plane, i.e. the line r R is not a mirror line.
Sec0ndar.v electron emission .from tungsten
185
total spectra for 0 = 50, 60, and 70" are not quite complete in the entire energy range
0 < E,, 6 Emax.When the combination of E,, and 8 is such that p , , / t i < K i I Mit, is seen
that only N-processes and two types of U-processes contribute, U(110) and U(221).
Obviously U(222) will be identical to the N-spectrum. In spite of the more complex geometry in the (111) case we thus only need to consider two U-process spectra in addition
to the N-process spectrum as long as plI is kept below Kl,, = 1.63 n/u. The spectrum for
8 = 0 will be the one-dimensional DOS along A(TP) (N) and along PH (U(110) U(211)).
4.2. (100)emission
The workfunction for electrons emitted from a (100) crystal plane of tungsten is 4s =
4.3 eV. This means that only that part of the band structure which is above E , + 4s 2:
1.23 Ryd above the muffin-tin zero (figure 5 of I and Christensen and Feuerbacher (1974))
will be available for SEE spectroscopy. It follows (figure 5 of paper I) that band 6 is the
lowest band that can contribute. In fact band 5 contributes only at very low kinetic
energies, i.e. in the energy regime where the experimental spectra are ill-defined due to
an instrumental 'spurious' peak (I). Therefore all calculations described in the following
include only the bands from number 6 and upwards (to 19).
The spectrum for 0 = 0 is just the one-dimensional DOS along the symmetry line A.
There are no U-processes that can introduce structure which is not contained in this
profile. Figure 6 shows the band structure along A and the calculated DOS together with
Figure 6 . The band structure of tungsten along (100) (A) as calculated, together with the
one-dimensional density-of-states function (DOS) and the experimental SEE spectrum (ezp)
of electrons emitted in the normal direction (0 = 0') from the (100) face.
the experimental trace for 8 = 0". The edges A and B of the band gap are defined by the
bands 6 and 7 and there is perfect agreement between the spectral positions of the band
edges in the experimental spectrum and the bulk band calculation. The lower edge is at
E , = 2.6 eV and the upper edge at EU = 4.7 eV (kinetic energy). Further, three peaks
(C, D, and E) in the experimental spectrum can be related directly to the DOS. The peak
C at E,, = 9.3 k 0.2 eV is due to emission from the minimum in band 7 at a A point
almost halfway between r a n d H. The experimental value, derived from SEE, of the energy
of this level is then 13.6 i 0.2 eV, which agrees with the calculation, 13.8 eV. The value of
01
186
N Egede Christensen and R F Willis
32.7 eV quoted by Smith et a1 (1976)differs by more than 1 eV from the calculation, and
the interpretation in Smith et al (1 976) concerning this spectral position does not seem
to be supported by the SEE experiment. Further comments on this element of structure are
given in 994.5 and 5.
The main peak D at Ekin= 10.1eV is due to emission from the almost triply degenerate level 14.4 eV above E , at r (bands number 7, 8 and 9). This assignment is made not
only from the agreement with calculated energy, but is supported by a cross-check with
other spectra ($4.5).A pronounced peak E at .Ekin= 18.0 eV coincides with a theoretical
DOS peak due to the critical point at H 22.3 eV above E , (bands 10 and 11).It is seen from
figure 6 that this last peak, which is at a very high energy, appears to be considerably
broader than the low-energy peaks where virtually no broadening is observed.
The off-normal spectra will be discussed according to the scheme presented in $ 3.1
and in $4.1. The K space plot for 0 = lo" is not shown due to the narrow pi, range. The
analysis of this spectrum follows the ones for larger 8 except for one particular peakthe lower band edge which appears as an extraordinary strong and sharp peak (cf.
figure 8 in I). The fact that this edge structure is particularly strong for 0 = 10" cannot
be accounted for immediately by the DCS calculation (see paper I).
The figures 7 to 12 show the spectra derived from the band structure together with
the experimental spectra (after background subtraction, see I). The use of these K space
diagrams follows from figures 3 and 4. The calculated spectra have not been broadened
(in paper I a Lorentzian broadening with 0.25 eV halfwidth was applied), and 'N'- and
important 'U'-contributions are shown separately. The main band gap, the gap between
H
Kl
K i n e t i c energy, E l e v )
Figure 7. K space piot, calculated DCS contributions ('Normal' and 'LJ(1,0,l)') and experimental SEE spectrum for emission from the (100) plane. The emission angle is 0 = 20".
The numbers at the K space root curves give the band indices. (The energy levels are counted
from below in energy.)
187
Secondary electron emission ,from tungsten
I " "
SEE
' " 1 ' i ' ' ' ' ' i '
" "
W (100) plane
+ = 00
1 " '
I " '
" '
"
'
"
0 =30°
Umklapp U ll,O,ll
\
I
Kinetic energy EleV)
KL
Figure 8. 'Normal' and 'U(101)'-contributions to the bulk
experimental spectrum (exp). (100) emission with 0 = 30".
I ' I ' I ' I
U 10.0.21
I ' " I
DCS
function together with the
l ' ' ' I ' l ' l " '
/
llOo1 plane
0=L0'
SEE
I
Umklapp
Umklapp U11.0.11
U11.011
I
I
v
IY I
-
1
1
Kll
@
Normal
Kinetic energy E ieV)
Kl
Figure 9. As figures 7 and 8, but for 0
=
40".
188
N Egede Christensen and R F Willis
4
Kinetic energy E levi
Figure 10. (100) emission, 0 = 50".The upper band gap edge B has its origin in band 7 at the
symmetry point N.
lane
+=00 8=60°
Kl
Figure 11. (100) emission, 0 = 60".
Kinetic energy E (eVI
189
Secondary electron emission from tungsten
KL
Kinetic energy E [ N I
Figure 12. (100) emission, 0 = 70".
band 6 and band 7, has in all cases its upper edge on the HNH line (G), and except for
8 = 70" this is also true for the lower edge. Thus, the gap edges map the bands 6 and 7 along
a G-line. This fact does not depend critically on the quantitative accuracy of the RAPW
band calculation but is rather a consequence of symmetry. The experiment gives Ekin
and pII,but since the symmetry ensures that the gap is on the HNH-line we can, from the
experimental pI,-values, derive the K vectors corresponding to the edges, and in this way
determine purely experimental dispersion relations for the electrons in band 6 and 7
along the line G. These points are marked in figure 25 (54.5). The band edges in the (100)
spectra are referred to as structure elements A and B in figures 7-12. It follows that a
peak S appears to move into the gap from the upper edge as 8 is increased. This peak
cannot be related to a peak in the bulk state density, and it is unambigiously verified
from figure 10 that S cannot be the upper edge itself. For 8 = 50" (figure 10)the calculated
upper edge is located at the symmetry point N. The energy of level 7 at N is 1.676 Ryd
over the muffin-tin zero or 10.8 eV above E,. This is consistent with the observation in
angle-resolved photoemission (Christensen and Feuerbacher 1974, Feuerbacher and
Christensen 1974)of the onset of bulk interband transitions from band 3 (at E,) to band 7
(near N), and we therefore know that the calculated spectral position of the upper edge for
6' = 50" is correct, i.e. B and not S is the upper band gap edge. Peak S is related to 'a
surface resonance ban8 (Willis et al 1977) (53.2). Some of the peaks (see figures 7-12)
appear to be related to the bulk spectra and have their main contributions at symmetry
lines. The origins of other peaks also appear from the figures. The peak D at E,, = 12.3 eV
in the spectrum for 8 = 30" (figure 8) is due to emission from band 9 at a K point on the
G-lines, and peak E is essentially due to emission from band 10. This emission has the
onset at E,, = 15.1 eV, at a A point halfway between r a n d H. This is the local minimum
of band 10 at this A point (see figure 6). The peak labelled D in the spectrum for 8 = 40"
190
N Eqede Christensen and R F Willis
(figure 9) has contributions from band 7 as well as band 8, ' N as well as U(101). The
actual calculation shows, however, that the intensity of the contributions from band 7
are dominating. The states from which this emission originates have their K vectors at
A points very close to TH/2, and the kinetic energy of D is 9.3 eV. This provides a new
check on the minimum of band 7 at TH/2 discussed in the case of 8 = 0".
It may appear, by inspection of the experimental spectra, that the linewidths of some
peaks are quite large. This cannot in all cases be taken as evidence for an increased
intrinsic broadening. Peak E in the 8 = 50" spectrum (figure 10) for example, appears to
be -2 eV wide. This does, however, agree with the theoretical N-process structure and it
is rather concluded that virtually no broadening is observed. Peak E has many contributions between E and E ' (figure 10). The trend which appeared to be the case with
8 = 0" showing an increased broadening with increasing energy (cf. peak E in figure 6)
does not hold in general.
4.3. (110)emission
As in the (100) case, the majority of the elements of structure observed in the SEE spectra
for emission from a (110) crystal plane can be related to the bulk band structure. As in
the previous case, there are, however, a few important exceptions.
The (110) spectrum for 8 = 0" is shown in figure 13 together with the one-dimensional
Kinetic
energy ieV)
Figure 13. (110) emission, 0 = 0". The curve shown with a full line is the one-dimensional
density-of-states function along the Z line ((llo), T-N). The curve shown with a broken
line is the experimental spectrum. The structure E is interpreted in terms of the VBRM mechanism (see text).
density of states along a C line. The experimental trace exhibits structure at 1.0 eV (A),
5.5 eV (R), 9.3 eV (C), 10.8 eV (D), 13.5 eV (E), 17.0 eV (F),and 17.8 eV (G). The lowenergy elements A and B are the main band edges whereas C is due to emission from the
bands 7, 8 and 9 at r. The workfunction for the (110) face is 4s = 5.1 eV, i.e. peak C is
14.4 eV above E , (cf. figure 5 of paper I). This agrees with the assignment of peak D in
the (100) spectrum for 8 = 0", where peak D was attributed to emission from these same
levels at r. The peak D in figure 13 is due to emission from the maximum of band 7 at
191
Secondary electron emission from tungsten
the C line (see figure 5 of I). The calculation shows that band 8 also has a maximum at
the X line, but the intensity is small since the extremum is very narrow. This explains
why it is not resolved in the experiment. The peak labelled E in figure 13 cannot be
related to structure in the DCS as derived from the present bulk band model. This cannot
be explained in terms of inaccuracies in the calculation since the band structurc has been
well verified for energies below as well as above the spectral positions of E. Similar
remarks may apply to F although this does not disagree as markedly from the DCS as
peak E. Further, the experimental peak Cis extraordinarily wide; it should rather be considered as consisting of two peaks C and C', with C' located at v 8.5 eV (kinetic energy).
The off-normal spectra for 8 = 20", 30", 40", 50°,60" and 70" are shown in the
figures 14-19. The band gap between band 6 and 7 is seen in all spectra although the
lower edge is ill-defined in a few of the experimental traces (8 = 40" and 6 = 60"). The
theoretical K space plots show that the band edges for small 6 are located at the NP
line. Again, this is a result of symmetry properties and is not very dependent on the
actual accuracy of the band calculation. Thus, for small 8, the experimental spectra give
information about energy as well as the K vector, and we have purely experimental points
of the dispersion relation E@).
The (110)spectrum for 8 = 20" which is shown in figure 14 contains nine elements of
structure which can all be interpreted in terms of the bulk band structure. They are, however, not all simply due to a high density of states alone. For example, the peak H is so
wide that it cannot be fully accounted for by the sharp peak in the calculated spectrum
at 18.5 eV to the edge at A (z6/16 x 27c/a) in band 11. This and the peak F are due to
SEE
c
sDectrum
\
a-
@-
Kl
Kinetic energy E lev)
Figure 14. (110) off-normal spectrum for 8 = 20". The calculated spectrum includes 'normal'
as well as 'Umklapp' contributions; the latter, however, are only important for energies above
35 eV, i.e. above the energy range covered by the experimental trace (top curve in right panel,
'exp'). Peak F is due to the VBRM transmission effect (shaded).
N Egede Christensen and R .F Willis
192
20
16
0
I
iu
...
......
._
.
.
.
....
.
L
0
0
16
12
2
10
20
30
LO
Kinetic energy E Id1
Kl
Figure 15. (110) emission for 0
0
L
=
30".
mechanism described in 43.2. As described in $3.2, the K points where the
condition may be fulfilled are determined by searching for points on the K space
curves where the slope equals the emission angle 8 (in this case 20"). Such points in the
bands 10 and 11 are marked in figure 14, and it is suggested that the peaks F and H are
due to enhanced emission from these states. Note that the sign of the normal component
of the group velocity is immaterial for these arguments since the (110) plane is a mirror
plane. The experimental spectrum contains, like the calculation, a minimum between
1123 and 12.9 eV with edges close to E and F. The calculation shows that the lower edge
of this gap is in band 9 at A (f4.S x 27c/a&) and the upper edge is in band 10 also at a
point (5.1 x & x 27cla) at the symmetry line A. The peaks C and D are due to emission
from the bands 8 and 9. The contributions, especially those to peak D, from band 9 are
not restricted to a particular point in K space since the K space curve (figure 14) of band
9 is almost parallel to the (110) direction over a large range of K,. Peak C has its main
contribution from the high density-of-states point at the NP line (band 8). The spectral
position and the width of the main gap (A-B) as observed (figure 14) agree well with the
calculation.
The peak G (16.0-16.8 eV) contains two strong DCS contributions from the bands 10
and 11, and the shoulder I may also be related to the calculated spectrum, but it follows
(figure 14) that it has contributions from very many bands over a large region in K space.
The lineshapes in the experimental spectrum for 8 = 30" (figure 15) differ somewhat
from those shown in figure 14 for 8 = 20". The peak D agrees in energy as well as width
with the corresponding peak in the calculated spectrum. It has contributions from the
the
VBRM
VBRM
193
Secondary electron emission from tungsten
Figure 16. (110) emission, 0 = 40". The calculated spectrum shows that 111 this case there
are gaps above the main gap (band 6 to 7). The extra gaps extend from 12.5 eV to 14.5 eV
and from 19.0 eV to 21.2 eV.
K
Kinetic energy E (ev]
Figurel'l. (110) emission, 8 = 50".The peaks D and F contain VBRM contributions (matching
at@ and @ or points symmetrical with respect to the I(,,axis).
194
N Egede Christensen and R F Willis
Secondary electron emission from tungsten
195
bands 7, 8 and 9. This is also the case for E, F and G, and it follows that essentially no
broadening effects are observed, in these cases, even for energies up to 19 eV above the
vacuum level. Well-definedenergy bands thus exist even 24 eV above the Fermi level. The
structure F due to band 10 is identical to the element E in the (100)spectrum for 8 = 30".
Peak C (figure 15) is considerably wider than the peak labelled C in the spectrum for
8 = 20". This can be explained by the VBRM mechanism (5 3.2). The spectrum for 8 = 40"
(figure 16) also shows extremely sharp structure even up to the highest energies ( E < hop
x 23.5 eV). This case is unique among the (110)spectra since the band calculation shows
that there are genuine gaps in addition to the usual 6-7 gap. These extra gaps extend
from 12.5 to 14.5 eV (bands 9 and 10) and from 19.0 to 21.2 eV (bands 11 and 12). The
edges are observed at (D, E) and (F,G) respectively. The sharp peak C is due to emission
from band 7 at the A point (almost) halfway between r and H. This is the critical point
(saddle point) in band 7 discussed previously (peak D in figure 9 and peak C in figure 6).
The upper edge B in the spectrum for 8 = 50" (figure 17) is well defined at Ekin= 4.0 eV
and agrees with the calculation. The lower edge, however, is observed at 2eV if it is
associated with the peak A. This is 1 eV below the predicted value. The calculation does
however, predict a peak below the edge itself (3 eV) due to emission either from band 6
at the NP line or a VBRM peak A (figure 17).The peak C has contributions from several
regions in K space. Its low-energy onset appears to be very close to the symmetry point
P in bands 8 and 9, and its high-energy part is due to N-emission from band 7. The width
of peak C can thus be fully accounted for in the calculated spectrum. The peak D contains on its high-energy side a strong, but narrow DCS contribution. The width of D
cannot be fully understood from the calculation. The peak F in the observed spectrum
is wide and is located at energies where the calculation predicts no structure. The elements G and H may be assigned to structure in the calculated spectrum as shown in the
figure. Peak H contains almost singular contributions from band 11 at the PN line (D)
as well as from band 12 at the A line. The strongest peak in the spectrum for 8 = 60"
at E = 12 eV is attributed (figure 18) to emission from a point close to the critical point
in band 9 at the symmetry line A (12.3 x & x 2n/a from r).A similar contribution is
observed as the peak D (also 12 eV) in the spectrum for 8 = 70" (figure 19).
4.4. Emissionfrom the (111)plane
In the previous two cases, (100) and (110) emission, it appeared that a very large number
of elements of structure in the SEE spectra could be related to structure in the DCS functions. Only a few peaks were ascribed to 'anomalies'. The analysis of the (111) spectra
does not quite follow this line. Although it will also appear that in this case several peaks
can be related to peaks in the DCS,all spectra contain several peaks which can only be
interpreted as 'anomalies'. Some of these peaks are even very strong and dominate the
spectra.
First the emission normal (6' = 0) to a (111) face is considered. The experimental
spectrum for 6' = 0" is shown in figure 20(b) together with the calculated one-dimensional
density-of-states function (which in this case is broadened by 0.2 eV). For comparison,
similar curves for the (100) case have been drawn in figure 2qa). These figures clearly
show the large difference between the correlation between SEE spectra and DCS functions
when the (100)and (111) case are compared. The (111) spectrum does contain an element
of structure, E, which relates to a peak in the density of states. This is assigned to emission from bands 7,8,9 at r, and a comparison with figure 20(a) shows that this interpretation is consistent with the analysis of the (100) spectrum. The peaks P and D are very
N Eqede Christensen and R F Willis
196
prominent features in the experimental spectrum and they cannot be related to the density
of states. Similar remarks apply to F which is located where the density-of-states function has a minimum. These three peaks are all interpreted as 'anomalies' of 'type (ivythe enhanced emission due to the particularly perfect matching between the bulk states
and the vacuum states ( V B R M 3.2).
~
The peak P at the kinetic energy 3 eV is very sharp
and is due to matching in band 6 just below the band gap at the A line. This is shown in
figure 21, where a part of the band structure is plotted along TP and PH (the continuation
of TP in the extended zone). It follows from the calculations (figures 20(b)and 21) that the
band gap between band 6 and band 7 which appears at the TP line is filled up by states
along PH. Therefore there is no genuine gap in the DCS function for 8 = 0". In spite of
lbl
10
20
30
Kinetic energy ieV 1
Figure 20. Comparison of normal-emission (0 =
0) spectra from the (100) plane (a) and the (111)
plane (b). The experimental spectra are shown
as curves with broken lines. Full curves are
calculated one-dimensional density-of-states
functions. The calculation in (b) includes the TP
as well as the PH lines.
l
r
I
1
l
AI
l
I
1
I
I
I
I
Fl
"
'
t
H
P
Figure 21. VBRM matching at the (111) symmetry line. Free-electron parabolas match the
and @ The peaks
band structure at @, @,
with the same labels in figure 20 are interpreted
as enhanced transmission due to the VBRM effect
with matching points as shown here.
0
197
Secondary electron emission from tungsten
this, it might be argued that lower gap edge at A could introduce structure, for example
via the mechanism referred to as 'anomalies of type (iii)' (0 2 and paper I). However, such
an interpretation would imply that the spectral position of the reduced-zone gap edge
in band 6 should be shifted by more than 1 eV. In contrast, the VBRM model predicts that
P should be located at Ekin= 3 eV in agreement with experiment. Further, the peak P
can be seen at all angles 8, and in all cases 0 d 8 6 70°, the VBRM does in fact predict the
correct spectral position. Thus P is not a lower-edge structure but due to an 'anomaly'
of type (iv) (VBRM),
except perhaps at 8 = 20" (cf. figure 17 of paper I). Figure 21 also
shows the matching points related to the peaks D and F. Peak D is due to matching in
the VBRM model in band 7 at a A point, whereas the matching point @ is in band 9 at
the PH-line (F).The high-energy peaks G and H at Ekin= 17 eV and 21 eV (figure 20)
can both be explained in terms of the DCS function which exhibits peaks at those energies.
The low-energy peak A may be related to the critical point in band 6 at H or a VBRM
peak close to this. The theoretical spectral positions of such peaks are EkinN 2.1 eV
which is somewhat above the experimental value of 2: 1.9 eV. However, the experimental
spectrum is somewhat ill-defined at these low energies (paper I) due to an 'instrumental'
peak at E,, = 0 which has been subtracted. The structure C bears resemblance to an
upper-edge contribution as observed in the (100) and (110) cases. The total DCS function
(broadened by 0.2 eV) in figure 20(b) does not contain prominent structure at the energy
of peak C. As mentioned earlier, there is no genuine gap, but there is a gap in the first BZ
alone (at the A line, figure 21). The minimum in band 7 at A gives a peak in the density
of states, but it is almost washed out by the broadening. The structure C may be this
DCS contribution or it may be a VBRM peak, the matching point being the one labelled
O i n figure 21.
The analysis of the off-normal spectra has been carried through for all angles
8 = lo", 20°, 30°, 40°, 50", 60" and 70" in the way described for the (100) and (110) cases.
Due to their complexity we will include here only two K space plots, and further we
have shown in these plots only the bands 6 to 9. Figure 22 shows the K space plot for 8 =
30". Note that this plot can only be applied up to E,, 2: 13 eV since band 9 is the uppermost band included. The experimental spectrum is shown with a broken line, and only
one calculated contribution (U(110))is shown. The N-processes consist of the two contributions 'Normal 1' and 'Normal 2' where the latter for convenience is shown in the
(222) zone, U(222) being identical to the N-process contribution. The 'Normal 1' and
'Normal 2' contributions are different because the (111) plane is not a mirror plane.
The total DCS function contains, in contrast to the DCS function for 0 = 0 , a finite
gap between the bands 6 and 7. This gap is defined by the edges at the points marked
and
and the gap is very small in energy. The U(110) partial DCS function has an
upper-edge peak at 6 eV which coincides with a clearly resolved peak (B) in the experimental trace. The experimental spectrum has further a rather broad peak, E, centred
around 13 eV which can be fully accounted for in the band calculation. The feature E
is related to the two close peaks E, and E, in the DCS function, which has edge contributions from E and E in band 9 in the (110) zone. Further, E, and E&have contributions
from severa other points,
and @ Although the experimental peak E appears to
be rather broad, virtually no roadenzng of the theoretical trace is needed to explain the
width. Also the structure F can be interpreted as a bulk DCS contribution. The shoulder
C is related to the edge-like feature in the DCS originating in band 8 at @in the first
Brillouin zone. The remaining peaks A, P and D are interpreted in terms of the VBRM
mechanism. The matching point corresponding to peak A is located at
in the zone
centred around (1, 1,0)27c/a. The particularly pronounced peak P has VBRM matching
0 8
Q Q
/A'I
N Egede Christensen and R F Willis
198
m,
a.
points at
and
The points
and /p31are in the ‘Normal 1’and ‘Normal 2’
regions of band 6, whereas P2 is in band 6 in the’zone centred at (2,4 1)27c/a.Of these
three matching points only P1 can give an essential contribution since this represents
the only state where the group velocity of the bulk state has a positive component towards the surface. The (111)plane is not a mirror plane, i.e. the rx line is not a mirror line,
and therefore there are no points corresponding to Ip2/and Ip3/ where the group velocity
has a component towards the surface. The central peak D is interpreted as a transmission
resonance in band 7, the matching point b e i n g n i n the (110) zone (figure 22). It follows
B
\
Figure 22. (111) emission, 6’ = 30” This plot can only be used up to E,,, ‘5 13 eV since band
9 is the uppermost band which is included. Only the U(110) contribution to the theoretical
DCS function is shown. The total spectrum has a very narrow band gap defined by the edges
at@ a n d o . Origins of DCS contributions to peaks are shown in circles. The points where the
VBRM condition is fulfilled are shown in squares. Special attention is paid to the peaks P and
D (see text).
that all elements of structure in the (111) spectrum for 0 = 30” have been accounted for,
R, C, E and F being assigned to high density-of-states regions, and A, P and D being
interpreted as transmission maxima due to the VBRM mechanism.
Figure 23 shows the K space plot for 6 = 40”. The experimental spectrum (broken
line) is shown together with the calculated ‘Normal’ and U(110) spectra. Again three
peaks A, P and D obviously cannot be related to the DCS function, but they can all, as in
the previous case, be interpreted as transmission resonances (VBRM). The essential contri-
Secondary electron emission from tungsten
199
bution to peak P is from the matching point /p1/ in the (110) zone. The 'Normal' DCS
function has very narrow peak at the same spectral position as peak P. However, since it
is extremely narrow it cannot be the origin of peak P which consequently is entirely
ascribed to the VBRM enhancement mechanism. The peaks B, C and E can all be related
to peaks in the DCS functions. The unlabelled structure at 19-20 eV does not correlate
with any of the two plotted DCS functions. It is located in the energy range where our
calculation is somewhat incomplete since it does not, for these energies, exhaust all Uprocesses. However, the detailed analysis showed that for this particular angle, there is a
VBRM peak at E,!, = 19.6 eV.
Figure 23. (111) emission, 0 = 40'. The calculated spectrum which is shown (full curve)
includes only 'normal' processes, i.e. emission from states in the first Brillouin zone.
matching points are shown in squares (see discussion in text).
VBRM
The analysis of the remaining (111) spectra follows essentially the examples discussed
above. The characteristic peak P is observed at all angles 0 = 0" to 8 = 70", and in all
cases its spectral position coincides with the predictions from the VBRM model to within
0.1 to 0.2 eV. The peak D is observed as 'extra' structure only for 8 = 0,10,20,30 and 40".
At larger angles it disappears and the maximum intensity is in a peak that correlates
with a theoretical DCS peak, namely the one corresponding to peak C in figures 22 and 23.
Also the calculations show that VBRM structure in the energy range 7.5 eV to 20 eV
disappears when 8 exceeds 40".
200
N Egede Christensen and R F Willis
-
This disappearance of peak D when 8 is increased above 40" can be seen in figure 24,
where the (111) spectra for 8 = 40" and 8 = 50" are compared. All theoretical VBRM
peaks are indicated by arrows in these figures. It is clearly seen that for both angles a
peak C correlates well with a main peak~inthe DCS function. The 40" spectrum contains
the peak D, exactly where theory predicts its spectral position. At 8 = 50" this peak is
absent in the experimental trace and simultaneously no VBRM peak corresponding to D
I
0
Ill1 plane1
L
8
12
16
20
Kinetic energy ieV1
Figure'%. Comparison of the (111) spectra for 6' = 40" and 0 = 50". The experimental spectra
are shown as full curves, whereas the curves shown with fine broken lines represent the
DCS functions. The arrows indicate theoretical spectral positions of VBRM peaks. All peaks
predicted by the VBRM model in the energy range 0-20 eV are indicated. Note the disappearance of peak D as H is increased from 40" to 50". Peak D is absent in all spectra for 0 > 40".
and this agrees with the prediction from the VBRM model (see also table 1).
is predicted. A rather pronounced shoulder at Ekin= 7.0 eV in the experimental trace
coincides with a 'new' VBRM peak at this energy, but in addition the shoulder may contain bulk DCS contributions. The VBRM structure at 19.6eV discussed earlier is also
indicated in figure 24. The spectral positions ofthe peaks ' P and ' D for various emission
angles 8 are listed in table 1, where they are compared with the values obtained from the
VBRM matching procedure.
The fact that VBRM effects are so prevalent on the (111)-face is not particular to tungsten, but should rather be considered as characteristic for BCC transition metals. The number of VBRM matching points depends essentially on the shape of the bands, which is a
consequence of the crystal structure. Therefore, we would expect also in the case of, for
instance, MO, to find many VBRM peaks in spectra from (111)faces and few in (100) and
(110) spectra. Of course, the energies will be different. A detailed investigation of the
201
Secondary electron emission from tungsten
Table 1. Spectral positions (kinetic energies) of the peaks ‘ P and ‘ D in (I 11) SEE spectra.
Emission
angle 0
(deg)
Peak ‘ P
experiment
(eV)
Peak ‘ P
theory
VBRM (eV)
Peak ‘D’
experiment
(ev)
Peak ‘D’
theory
VBRM (eV)
0
10
20
30
40
50
60
70
3.0
3.0
3.4
3.2
3.5
3.5
3.5
3.3
3.0
3.2
3.4
3.2
3.6
3.4, 3.5
3.5
3.4
1.5
1.6
7.5
5.2
9.1
1.5
1.4
747.1
5.2
9.0
occurrence of VBRM points in other structures has not yet been made. It can be mentioned,
however, that in the case of silver (FCC), there is only one such matching point in a ‘relevant’
energy range along the (111) symmetry line.
4.5. ‘Experimental’ band structure of tungsten
The analysis of the experimental spectra in $9 4.2-4.4 demonstrates that virtually all
elements of structure in the SEE spectra (Willis and Christensen 1978) can be accounted
for. A large number of peaks are related to high density-of-states contributions, essentially at edges in the K space slice scanned. These edges define either genuine gap edges or
edges in bands at energies where other bands overlap. Further, several peaks are assigned
to ‘anomalies’, a surface resonance band and the enhancement due to the VBRM mechanism (9 3.2).
Here we shall briefly summarise some of the assignments of the elements of structure
that earlier were interpreted as DCS structure, and in particular pay attention to those
peaks that have their origin in K space at symmetry lines. These are particularly interesting, since the locations in K space for many of these structure elements do not depend
sensitively on the actual accuracy of the band calculation. The fact that the K space
origins lie on the symmetry lines is a consequence of symmetry and the shape of the
bands, i.e. we know that these points are on symmetry lines and we need not use the band
calculation to determine the K , component of the wavevector. For these particular
points, the experiment gives energy as well as KIland K,, i.e. we obtain purely experimental
points for the dispersion relation of the electrons in the crystal.
The calculated (RAPW) band structure along symmetry lines of tungsten is shown in
figure 25, and in the same figure are shown 34 different ‘experimental’ points. These
points have been numbered 1-34, and table 2 serves as entry to the figures in the previous
sections where the label and emission angle are given. It follows from figure 25 that the
band calculation agrees with the experiment over the entire range from the vacuum levels
up to N 24 eV above the Fermi level. The estimated maximum error of reading from the
experimental curves is estimated to f0.2 eV. Therefore, although the 34 points in figure
25 represent only a part of the experimental elements of structure which have been interpreted, they are sufficient to demonstrate the applicability of angle-resolved secondary
electron emission for spectroscopy of one-electron states in tungsten. This conclusion,
202
N Egede Christensen and R F Willis
Figure 25. Band structure of tungsten along symmetry lines as derived by the RAPW method
(potential V,, (Christensen and Feuerbacher 1974)). The points numbered 1-34 are deduced
from the analysis of the experimental SEE spectra. These points, which are located at symmetry
lines, represent experimental points in the band structure since, in these cases, the experiment
gives energy K ,,,as well as K, (see text). (The numbers refer to table 2.)
drawn from the result displayed in figure 25, is further supported by the comparison of
experimental and theoretical spectral positions of the band gap edges in the (100) and
(110) spectra as shown in paper I.
As follows from table 2, and from &4.2-4.4,some critical points of the band structure
are observed in several different spectra. For example, point 1 at r is seen, as should be
expected, in all three normal-emission (6 = 0) spectra. Also the point 2, the minimum in
band 7, halfway between r and H at the A line, has been observed in three different
spectra.
203
Secondary electron emission from tungsten
Table 2. Identification of the points marked in figure 25. (Comparison between ‘experimentay
band structure and theoretical band structure of tungsten.)
Point number
in figure 25
Observed in
emission from
the plane
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Emission angle
(ded
0
0
0
0
40
40
50
70
60
30
30
0
20
50
0
0
10
10
20
20
30
30
40
40
50
50
60
60
70
50
30
50
30
50
20
30
30
20
50
Label
In figure
6
13
20b
6
9
16
10
E
F
E
H
H
A
B
A
B
A
B
A
B
A
B
A
B
A
B
B
D
D
C
G
H
B
B
A
A
G
19
16
8
15
6
14
17
6
6
I
7
8
8
9
9
10
10
11
11
12
10
8
17
15
17
14
15
15
14
17
5. Summary and conclusions
Angle-resolved secondary electron emission spectroscopy constitutes an important tool
for investigation of the electronic structure of single crystals. A large number of elements
of structure in the SEE spectra have been interpreted in terms of density-of-contributingstates functions (DCS) derived from a band structure calculation for the bulk material.
The majority of the peaks in the SEE spectra which can be related to peaks in the DCS
functions have their origin at edges of the K space curves corresponding to the various
204
N Egede Christensen and R F Willis
bands, i.e. points where aEMli;k: = 0 in (19). This explains why the DCS peaks correlate
with peaks in the SEE spectra although the SEE current in the angle-resolved measurements is expected to bear only a superficial resemblance to the actual bulk and surface
local density-of-states. Several elements of structure have their origin at symmetry lines,
which implies that, assuming conservation of parallel momentum, the experiment gives
the energy as well as the total wavevector of the emitting states. As a result we obtain an
'experimental band structure' of tungsten. This agrees with the calculation (RAPW).
The agreement between the experimental and theoretical band structure is accurate to
within the error of reading of the experimental traces N k (0.1-0.2 eV) in the entire energy
range from the vacuum levels up to N 24 eV above the Fermi level. This close agreement
may seem surprising for the following reason. The RAPW band structure is not selfconsistent. The crystal potential does contain semi-empirical parameters (Christensen
and Feuerbacher 1978) such as exchange weight and choice of electron configuration in
the basic atomic calculation. The choice of the actual potential function was essentially
based on the agreement between experiment and theory in an energy range close to the
Fermi level. The present work uses a band model which is an extension to higher energies
of'the band structure of Christensen and Feuerbacher (1974),i.e. based on the same potential V,. The fact that the band model agrees with experiments involving small excitation
energies, such as in Fermi surface experiments implies that many-body corrections, if
essential, effectively must have been folded into the ad hoc one-electron potential function. The self-energy corrections are largest in magnitude and negative (Lundqvist 1969)
in the low-energy regime and decrease in magnitude with increasing energy. Therefore,
the application of the same potential in the high-energy regime as the one that seems to contain sufficient corrections near E, can be expected to lead to theoretical energy bands which
are too low in energy. Such predictions do not agree with the present observations,
however. From this it might be concluded that many-body corrections are in fact very
small in the entire energy range E , < E d E , + E , En z 25eV. A more probable
explanation follows from the observation that the high-lying energy bands are considerably less sensitive to variations in the crystal potential function than the states
near E , (Christensen and Feuerbacher J 974). Thus a crystal potential which is somewhat
incorrect in the high-energy regime may still yield reasonable energy eigenvalues.
A somewhat less critical, but independent, test of the band calculation in the highenergy regime is provided by a comparison'with optical reflection experiments from which
the imaginary part e 2 ( o )of the dielectric function may be derived. In general, this function
does not immediately give information about critical-point transitions as these may be
masked (Christensen and Seraphin 1971) by a large non-critical bakkground. The
comparison in figure 26 between eZ(w) as derived from the experiments by Weaver et al
(1975) and €,(CO) calculated from the RAPW band structure therefore serves as an over-all
check. The theoretical €,-profile is just the joint density-of-states function divided by
(hw),, i.e. we have set all matrix elements constant. The calculated spectrum has been
scaled to the experimental at ho = 11.2 eV, and no further adjustment is made. In view
of the crude approximation concerning the transition matrix elements, the agreement
between theory and experiment (figure 26) is remarkably good. The slope of the E ,
curve from 11 to 14.5eV is reproduced by the calculation, and the broad maximum
around ho = 16.5 eV can be related to structure in the theoretical curve. Since the optical
experiment (Weaver et al. 1975) is assumed essentially to monitor bulk electronic properties it follows from the comparison in figure 26 that the present band model represents
the electronic structure in the bulk.
In the case of angle-resolved photoemission (Christensen and Feuerbacher 1974,
205
Secondary electron emission from tungsten
Feuerbacher and Christensen 1974), it was found that the band structure was modified
when passing from the bulk towards the surface, where the d bands are narrowed.
The present analysis of the SEE spectra shows that such modifications are not to be found
in the high-energy regime, i.e. within the error 5 0 . 2 eV we find that the band structure
close to the surface is undistorted from the bulk band structure. This does not mean that
there are no specific surface electronic structure effects to be observed. We have in the
present work observed a number of ‘anomalies’, i.e. contributions to the SEE spectra which
cannot be related directly to the bulk band structure, but which are surface-specific
properties.
t
€2
W
-
3-
2-
1-
10
12
11
16
18
20
Photon energy, h w ( e V )
22
2L
Figure 26. Imaginary part ~ ~ (
of 0
the)dielectric function for ho > 10 eV. The experimental
curve was obtained by Weaver et al(1975). The broken curve is derived from the joint density
of states calculated from the RAPW band model.
The features which have been referred to as ‘anomalies’ include the surface resonance
band on the (100) face, emission in band gaps even when no surface resonance exists,
and particularly strong emission from states where the (bulk) band structure matches
the dispersion relation of the emitted electrons in vacuum. Emission at energies falling in
a band gap is explained as emission via vacuum wavefunctions tailing into the crystal.
This, together with emission from the surface state, represents ‘surface emission’ which
has also been observed in angle-resolved photoemission.
Secondary electron emission spectroscopy is an important supplement to photoemission. SEE allows the study of the states above the vacuum level in a way that is free
of the complications from the optical transition matrix elements. These matrix elements
constitute a major difficulty in photoemission models since they require an evaluation
of the electromagnetic field in the solid. Accurate calculations must therefore include
evaluation of the transverse as well as the longitudinal dielectric functions (Hasselberg
1976, Kliewer 1976). Such calculations themselves require a detailed knowledge of the
electronic properties and optical matrix elements. Therefore a complicated self-consistent
calculation is necessary.
The states which are monitored by SEE spectroscopy are essentially the final states
involved in photoemission. There may, however, be cases where the energies observed
in SEE differ from those detected by photoemission. The reason is that in photoemission,
where the elastic primaries are studied, if the hole which is created by the excitation is
206
N Eqede Christensen and R F Willis
localised, it introduces relaxation shifts of the final states. As an example of one such
possible difference between final-state energies measured by SEE and PE,one can consider
the point labelled 2 in figure 25, the minimum of band 7 halfway between r and H on
the A line. The calculated energy of this level is 13.8 eV above E , and this agrees with
SEE. In contrast the analysis of constant initial state (CIS) photoemission spectra by Smith
et al(1976) suggests that the same level is 12.7 eV above E,, i.e. there is an apparent discrepancy of more than 1 eV which exceeds the inaccuracy of our analysis.
Apart from being a supplement to photoemission spectroscopy, the present analysis
has repercussions for the methods of interpretation of PE experiments. The ‘anomalies’
which we have observed are of general character and they will also influence photoemission spectra. The CIS spectra (figure 27) measured by Smith et a1 (1976) clearly show
Figure 27. Photoemission from a (100) face of tungsten in the normal direction (Smith et al
1976). The spectrum CIS (constant initial state) was obtained by varying the photon energy
and analyser energy (kinetic energy) synchronously. The emission in the band gap is shown
as the hatched area.
that there is a substantial emission (hatched in figure 27) in the band gap, i.e. the ‘surface
emission’ effect. Further, it follows from our analysis in $2, that emission via states
tailing into the solid is not restricted to energy ranges corresponding to band gaps,
‘Surface emission’ may often, however, be masked by intense bulk direct contributions
to the photoemission spectra, when such processes are possible. The photoemission
spectra from the (110) faces of the noble metals measured by Heiman et a1 (1976a, b)
showed that these spectra can be related to the one-dimensional density of initial states
alone, in spite of the fact that direct transitions are energetically possible. This can be
explained as emission following optical excitation from initial states to final states which
tail a very short distance into the solid. The reason why direct transitions are suppressed
in these measurements, and therefore cannot mask the surface emission, is due to the
almost grazing incidence (Heimann et a1 1976a, b) of the light beam with respect to the
surface and the final states which are energetically accessible having a low content of a
plane-wave component corresponding to the (110) reciprocal lattice vector. This reciprocal lattice vector is that appropriate for ‘Gvector supported’ direct bulk transitions
(Mahan 1970,Willias and Feuerbacher 1977).The ‘transmission anomalies’ which we have
observed in the SEE spectra are quite general phenomena, and they will therefore also
Secondary electron emission from tungsten
207
influence photoemission spectra. Such effects can easily be confused with optical matrix
element effects or intrinsic density of states effects, whereas in reality they constitute
a complex ‘final-state modulation effect’ on the PE spectra.? There is one important
proviso, however. In photoemission, the optical matrix element selects specific final
states. The conditions for ‘transmission anomalies’ in the elastic photoemission current
are therefore more stringent and likely to be less occurrent for any particular fixed photon
excitation energy. Recent photoemission wave-matching calculations (Spanjaard et al
1977) appear to support this view.
Acknowledgments
The authors wish to express their thanks to Dr B Feuerbacher for many stimulating
discussions. The numerical calculations were carried out at the Northern European
University Computing Center (NEUCC), Lyngby, Denmark. The staff are thanked for
the excellent service during the period of this work.
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~
t This would support, to some extent, the very early group velocity arguments against the interpretation of
photoemission spectra solely in terms of density-of-states features (E 0 Kane 1966 J . Phys. Soc. Japan 21
(Suppl) 37). However, these arguments were based on electron transport effects only, and ignored the important
consequences of wave matching at the surface.