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Photofield emission spectroscopy of surface electronic states of tungsten
Surface Science 172 (1986) 477-495 North-Holland, Amsterdam
PHOTOFIELD EMISSION SPECTROSCOPY ELECTRONIC STATES OF TUNGSTEN
477
OF SURFACE
David VENUS Department of Physics, University of Toronto, Toronto, Ontario M5S IA 7, Canada
and M a r t i n J.G. L E E Department of Physics and Scarborough College, University of Toronto, Toronto, Ontario M5S IA 7, Canada
Received 29 July 1985; accepted for publication 29 January 1986 A novel photofield emission spectrometer has been used to observe departures from free electron behaviour in the total energy distributions of photofield emission from the (100), (110), (111), (211), and (510) facets of a tungsten field emitter. Measurements with p-polarized light at grazing incidence at seven photon energies in the visible and near ultraviolet show departures from free electron behaviour having the same form as those observed in field emission distributions, but shifted to higher energy by one photon energy. The shapes and polarization dependence of the distributions are precisely those expected for surface photoexcitation from electron states near the surface to vacuum tunnelling final states. These observations provide strong evidence that the surface photoeffect is the dominant mechanism of excitation by p-polarized light in photofield emission from tungsten.
I. Introduction P h o t o f i e l d emission is the p h e n o m e n o n w h e r e b y p h o t o e x c i t e d electrons escape f r o m a solid b y tunnelling to the v a c u u m in the presence of a large electric field a p p l i e d to the surface. It has features in c o m m o n with b o t h p h o t o e m i s s i o n a n d field emission. W h e r e a s field emission p r o b e s electronic states b e l o w the F e r m i level b y using large surface electric fields to allow tunnelling, a n d p h o t o e m i s s i o n p r o b e s u n b o u n d electronic states a b o v e the v a c u u m level b y using p h o t o e x c i t a t i o n , p h o t o f i e l d emission explores the i n t e r m e d i a t e energy range b e t w e e n the F e r m i level a n d the v a c u u m level. Electronic excitation in p h o t o f i e l d emission has been a t t r i b u t e d to the surface p h o t o e l e c t r i c effect [1]. This i n t e r p r e t a t i o n has been inferred f r o m the field d e p e n d e n c e o f the total p h o t o c u r r e n t [2], a n d b y fitting [3,4] the d i s t r i b u tions to free electron m o d e l s of p h o t o f i e l d emission which a s s u m e surface p h o t o e x c i t a t i o n [5,6]. M e a s u r e m e n t s of the p o l a r i z a t i o n d e p e n d e n c e of the p h o t o c u r r e n t [7,8] have also i n d i c a t e d that the d o m i n a n t m e c h a n i s m o f 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
478
D, Venus, M.J.G. Lee / PES of surface electronic states of W
excitation is the surface photoelectric effect. This has led to the expectation that photofield emission is a very sensitive probe of electronic states at the surface of a metal, and that it may be useful in studies of the surface potential barrier, the detection of unoccupied surface states between the Fermi level and the vacuum level, the investigation of the nonlocal electromagnetic response of a metal surface, and studies of adsorbate systems. The dependence of the total energy distribution on the photon energy can serve as a further test of the hypothesis that surface excitation is the dominant photoexcitation mechanism in photofield emission. Field emission experiments have revealed significant departures from free electron behaviour in the surface density of states at the initial state energy [9,10]. The surface photoexcitation hypothesis implies that these same departures will also be observed in photofield emission distributions, but shifted to higher energy by the energy of one photon. The identification of the mechanism of excitation in photofield emission will not be conclusive until either such structure is observed or its absence is accounted for. Recent experimental findings of Gao and Reifenberger [11] have gone some way to addressing this question. They have measured the photofield emission total energy distribution from the (100) plane of tungsten with light of wavelength 477 nm, and report a large secondary peak in the distribution. This peak is attributed to surface photoexcitation from the well-known surface resonance (the "Swanson hump") on this plane 0.35 eV below the Fermi energy [12]. However, as their published data are restricted to a single photon energy, the dependence of the distribution on photon energy cannot be evaluated. In the present paper, photo field emission distributions are reported on the (100), (110), (111), (211), and (510) crystallographic planes of a tungsten field emitter, using p-polarized light at grazing incidence at seven photon energies ranging from 1.916 to 3.536 eV. The distributions measured on the (100) plane are consistent with that of Gao and Reifenberger, and many departures from free electron behaviour on other planes are reported for the first time in photofield emission. At each photon energy, departures from free electron behaviour in the photofield emission distributions are found to be in one-to-one correspondence with the initial state structure observed in field emission, providing conclusive evidence of surface excitation under these polarization conditions. This paper is divided into four sections. In section 2 the theory of surface excitation is reviewed. In section 3 the experimental procedure and results are described. Finally, the conclusions of this work are summarized in section 4. 2. Theory When a strong static electric field is applied to the surface of a metal, the surface potential changes from a step to a roughly triangular barrier, through
D. Venus, M.J.G. Lee / PES of surface electronicstates of W
479
which electrons may tunnel to the vacuum region. To observe the tunnelling of conduction electrons through the surface barrier ("field emission") from tungsten, an electric field of the order of 0.2 V/,~ is required. In order to obtain such a large field, the sample is chosen to be a thin wire, the end of which has been electrochemically etched to form a cone with a hemispherical endcap of radius --- 0.1/xm, called the "tip". Because of its small size the tip is a single crystal, with facets approximately 500 ,~ square normal to the low index crystallographic directions. The sample is spot welded to a tungsten support loop, and mounted in an ultra-high vacuum chamber operating in the 10-11 Torr range. A potential difference of = 2000 V applied between the tip and a conducting fluorescent screen creates the required field at the tip surface. The field emission current causes fluorescence on the screen, and a characteristic pattern is produced by the variations in the work function from one tip facet to another. The appearance of the field emission pattern indicates the state of cleanliness of the sample. The sample is cleaned whenever necessary by electrically heating the support loop to incandescence. Field emission from a single crystallographic direction may be selected by locating a small probe hole in the screen. The probe hole current can be analysed using appropriate electron optics to yield the total energy distribution of the electrons. In photofield emission, laser light with a photon energy less than the work function of the metal is focussed on the tip, and the total energy distribution of the photofield emission is measured. The Hamiltonian which describes the system of electrons is H o = - (h2/2m)V
(1)
2 + V(r) + H',
where V ( r ) = lie(,'),
Z < 0,
V(r)=V~(z)=-eFz-e2/az,
z>O.
The crystal potential Vc(r ) is the three-dimensional periodic potential inside a metal. The surface potential Vs(z ) has terms resulting from the applied electric field F and the image charge, and varies only in the direction of the surface normal, L The perturbing electromagnetic dipole interaction which causes photoexcitation is given by H ' = -ihe2m____c[ V . A ( r )
+A(r).
V] e -i*'t,
(2)
where - i h ~ 7 is the electron momentum and the monochromatic electromagnetic vector potential A (r)exp(-i~0t) is written in the gauge where the scalar potential is identically zero. Inside the metal, the eigenstates [M>, ( M stands for a set of quantum numbers) are solutions of the SchriSdinger equation for the crystal potential. In the vacuum region, the eigenstates are a product of free electron states of
D. Venus, M.J.G. Lee / PES of surface electronic states of W
480
transverse wavevector p parallel to the interface and complicated tunnelling functions in the z-direction. The tunnelhng function of energy E M, transverse momentum p, and unit amplitude at the surface, is well approximated far from the surface by a plane wave whose amplitude is equal to the tunnelling coefficient T(W~). Since T(W~) is an exponential function [13] of the normal energy
W~, = EM -- ( h2/2,, ) p 2, the majority of the emission transverse wavevector. Following the field emission the photoemission calculation emission current excited from written as [16] 2~re
JL = - - ~ - - E S ( E L + h ° ° - - E M )
(3) current will derive from states with small calculation of Nicolaou and Modinos [14] and of Schaich and Ashcroft [15], the photofield the initially unperturbed state L ) may be
IDMLI 2 E I C ~ l 2 IT(WMp) I 2
M
(4)
p
In this expression the optical transition matrix element DML is given by
DML =
ihe 2mc(MIV.A(r)+A(r).VIL),
(5)
and the delta function allows transitions to final states I M ) only if the transition conserves energy. The C~p are the expansion coefficients required at the interface (z = 0) to match the crystal state of energy E M and transverse momentum k M to the tunnelling eigenfunction of energy E M and transverse momentum p which propagates in the the + z direction. The sum includes all those transverse momenta which satisfy the condition
P:kM+g,
(6)
where g is a surface reciprocal lattice vector. This matching condition implies that transmission through a plane surface conserves the transverse crystal momentum only to within a surface reciprocal lattice vector. The total energy distribution of the photofield emission current from all crystal states is
J ( E ) = ~'~f(EL) JL [ 1 - - f ( E M ) ] 8 ( E - E M ) , L
(7)
where f ( E ) is the Fermi-Dirac distribution function. The matrix element DML describes the photoexcitation step in photofield emission. It is instructive to rewrite the matrix element in the form
DML = f l ( M I [ V . A ( r ) ]
[L) + 2 f l ( M l A ( r ) . ~7IL),
(8)
where fl = - i h e / 2 m c . The first term in this equation represents transitions induced by a spatial variation of the vector potential. If the incident light has a
D. Venus, M.J.G. Lee / PES of surface electronic states of W
481
component of vector potential normal to the plane metal-vacuum interface, the vector potential will vary rapidly in the interface region and cause this term to contribute to surface photoexcitation. In a simple local response theory, this rapid variation takes the form of a discontinuity in the component of the vector potential normal to the interface, while the component of the vector potential parallel to the interface is conserved and does not excite electronic transitions at the surface. This contribution to the surface excitation is essentially unchanged if a more physically correct nonlocal response theory for the surface is considered [17]. An additional effect is incorporated in the nonlocal calculation of Feibelman [18], who uses the self-consistent surface potential calculated by Lang and Kohn [19]. In this potential, the metal surface is defined by the sharp boundary of the positively charged background "jellium". The electron density is no longer confined to the z < 0 side of the interface, but spills out into the vacuum region very close to the surface in a sigmoid distribution. Feibelman demonstrates that at frequencies below the bulk plasma frequency, the normal component of the vector potential has a large peak in the surface region which greatly "overshoots" the change expected from the local theory. This peak decays to the local theory value via Friedel-like oscillations. The peak may be understood qualitatively [20] as a plasma oscillation localized within the region of low electron density at z > 0. Photoemission experiments by Levinson and Plummer [21] have verified that the surface peak in A z due to this effect makes a significant contribution to the surface photoexcitation for aluminum. This effect is also expected to be important in photofield emission, since the optical frequencies used in photofield emission are typically much smaller than the plasma frequency. The second term in the matrix element in eq. (8) can also make a contribution to surface photoexcitation, but through a slightly different mechanism. Even if the vector potential has zero divergence, this term will be non-zero. Under these circumstances, it may be evaluated using the commutator identity [1]
2flA • ~ M I W I L) =- ~ A .
~MIITV(r) I L).
(9)
In the surface region the potential V(r) is given by the surface potential V~(z), and this term results in the surface photoelectric effect in its original, and narrowest, definition. Since the surface potential varies only in the normal direction, only the normal component of the vector potential is effective in exciting these transitions. Thus the surface excitation described by both terms in eq. (8) will be maximized by choosing the polarization vector of the incident light to be normal to the surface. This may be accomplished by using p-polarized light at grazing incidence. Under these conditions, previous studies [8] have shown that approximately 0.5% of the total photofield emission
D. Venus, M.J.G. Lee / PES of surface electronic states" of W
482
current has a polarization dependence that is inconsistent with surface excitation. Field emission and photofield emission are sensitive to surface electronic states because the matching coefficients C~p in eq. (4) weight the emission current approximately as the surface density of states at the final state energy. In photofield emission due to surface excitation, the photocurrent further depends on the surface electronic states because the matrix elements DML are appreciable only close to the surface. Since surface optical transitions need not conserve the component of electron m o m e n t u m normal to the interface, and since only states with small transverse wavevector contribute significantly to the emitted current, the matrix elements weight the current approximately as the product of the initial and final surface densities of states at k = 0. Together, these effects are expected to produce departures from free electron behaviour in the total energy distributions. The transitions will involve the same initial states as in field emission. Therefore, those departures from free electron behaviour observed in field emission energy distributions [9,10] are expected to appear also in the photofield emission distributions, but shifted to higher energy by the energy of one photon. It is not clear, a priori, whether the final density of states will be governed principally by the distribution of electron states at the surface of the crystal or in the vacuum tunnelling region. Experimental studies of the total energy distribution may help to resolve this question. In order to identify departures from free electron behaviour due to surface photoexcitation, it is useful to reduce eq. (4) to the limit of the free electron model. The potential in a free electron metal is not periodic, so the sum over p collapses to the single term p = k M. Since the crystal states are free electron states, C~te is unity, and the sums over L and M represent integrals over total wavevector states K L and K M. The matrix elements involve only functions of the spatial coordinate z, and may be written as
I DML 12 =
(47r2/ct)
lAMa [2 3(kM-- kL),
(10)
where AML is the matrix d e m e n t between the one-dimensional wavefunctions in the z-direction, and ct is the area of the interface. These integrals may be transformed to integrals over total energy, normal energy, and azimuthal angle, and evaluated to yield an expression for the total energy distribution at the detector,
J(E) = 4m2e122/(Ir3hSct) f ( E - h~) xf E
[1 - f ( E ) ]
-1/2 laMLt 2 Ir(W)l
(11)
" Vo+hw
in agreement with the results of Schwartz and Cole [6]. In this equation, 12 is the volume within which the wavefunctions are normalized, and V0 is the
D. Venus, M.J.G. Lee / PES of surface electronic states of W
483
energy at the bottom of the free electron conduction band. The barrier tunnelling probability may be evaluated in the WKB approximation and expanded about E F + h~ as [13] IT(W) 12 = e -c e t w - e ~ - h ~ ' ) / a ,
(12)
where c = 4 ( 2 m ~ , * ) a / 2 v ( y ) , d = h e F / [ 2 ( 2 m e p * ) a / 2 t ( y ) ] , y = ( e 3 F ) l / 2 / e p *, ~* = ]~-h~0 I, ~ is the work function and v ( y ) and t ( y ) are elliptic integrals defined in ref. [13]. Since the factor [ ( W - V o ) ( W - h c o - V 0 ) ] -1/2 varies slowly compared to the exponential, it may be approximated by its endpoint. If the matrix element is assumed to be a constant, ]A[ 2, for all initial and final states separated by one photon energy, then eq. (11) yields 8mf22 [ 1 - f ( E ) ]
J(E)-
[AI2 [ ( E - V o ) ( E - h ~ -
Vo)] -'/2 J~-E(E),
•/.r h 2o~
(13a) where J~E(E)
reed
-
-
.
2--~3/(e-
h~) IT(E)
[2
(13b)
reduces to the field emission current density [22] evaluated with an effective Fermi level E F + h w. As the factor [ ( E - V o ) ( E - h w - ~)]-1/2 is slowly varying in comparison with J~E(E), the total energy distribution in photofield emission from a free electron metal is given to a good approximation by shifting the field emission distribution by the photon energy h0~. A plot of the logarithm of the current against energy is roughly triangular in shape. The constant positive slope at low energy is 1 / d , and depends only on the barrier transmission. The constant negative slope at high energy is 1 / d - 1 / k T , and results from the barrier transmission and the cutoff of the Fermi occupation factor. Extrapolations of these constant slope regions intersect at E F + hw. Departures from free electron behaviour will therefore appear as changes in slope on the linear portions of the logarithmic plot.
3. Experiment and results The photofield emission spectrometer used to measure the total energy distributions has been described elsewhere [23]. This apparatus has two important features that were not available in the apparatus used in previous studies. The first of these features is a sample positioner which allows the laser light to fall on any crystallographic facet of the field emitter with any desired angles of incidence and linear polarization. If the facet is illuminated by p-polarized light at grazing incidence, the photocurrent due to surface excita-
484
D. Venus, M.J.G. Lee / PES of surface electronic states of W
tion is expected to dominate the total photocurrent. The second feature is a deflection energy analyser consisting of two 127 ° cylindrical analysers in tandem. This analyser has a resolution function with a full width at half m a x i m u m height of (0.0390 + 0.0006)E 0, where E 0 is the pass kinetic energy of the analyser, and a peak signal-to-noise ratio of 2 × 10 4. The large signalto-noise ratio makes it possible to extend measurements of the total energy distribution over four orders of magnitude. By plotting the logarithm of the distribution against energy, it is possible to detect departures from free electron behaviour. Total energy distributions have been measured for five well defined regions of the tungsten field emission pattern, corresponding to the (100), (110), (111), (211) and (510) crystallographic directions. The light source was a krypton ion laser which operates at seven wavelengths in the visible and near ultraviolet corresponding to photon energies between 1.916 and 3.536 eV. Low count rates from planes with large work functions, such as the (110), made measurements at the lower photon energies very difficult. These data are not included since they have too much shot noise to be of use. Field emission and photofield emission distributions were recorded by means of a computer controlled data acquisition system. The pass energy of the electron energy analyser was set to 2 eV, giving an instrumental function with a full width at half maximum height of approximately 80 mV. The distribution was divided into about one hundred equal intervals, or channels, each having a width of approximately 24 mV. The entire distribution was collected every 200 ms by sweeping through the channels sequentially and recording the counts accumulated in each channel in 2 ms. The results from each sweep were added to average out the effects of alignment drift and slow tip contamination. Data were accumulated for between 15 and 30 min, depending on the size of the statistical noise. A F o w l e r - N o r d h e i m plot was taken to establish the field constant of the tip for each facet. TypicallY,othe field at which the distributions were measured was within 10% of 0.25 V / A . In order to resolve departures from free electron behaviour in the total energy distribution, it is necessary to remove the influence of the barrier transmission probability by subtracting the free electron "background". According to eq. (13), the free electron contribution to the photofield emission distribution may be parameterized as In J ( E ) = l n In J ( E )
A-(E-E
v-h~o)m
1,
= In A - ( E - E F - h r o ) m 2,
-5kT,
(14a)
E - E v - ho~ > 5 k T ,
(14b)
E-E
v-boo<
where A is the amplitude of the distribution and the slopes rn~ and m 2 may be related to the temperature and applied field using eq. (13). If straight lines are fitted to the linear portion of the logarithm of the distribution at high and low energy, the slopes and intercepts of these lines determine the four parameters A , E v, m~ and m 2. Such fits have been made to the data measured on all five
D. Venus, M.J.G. Lee / PES of surface electronic states of W
485
planes, and yield a consistent value of the Fermi energy with an uncertainty of + 17 meV. To bring the departures from free electron behaviour into greater prominence, the enhancement factor R ( E ) has been calculated as the difference between the natural logarithms of the experimental and fitted free electron curves. R(E)
= In J e x p ( E ) - In Jfree(E).
(15)
The energy range in which the enhancement factor contains useful information is restricted to the linear portion of the low energy side of the fitted distribution. The linear portion of the high energy side of the distribution is so steep that deviations from linearity cannot be reliably detected. The rounded portion at the apex of the fitted distribution is the energy range where the Fermi function varies most rapidly. Structure in this region of the enhancement factor is sensitive to small changes in the fit and is not reproducible. At very low energies, the enhancement factor becomes large because the field emission distribution overlaps the photofield emission distribution. Enhancement factors for the data sets measured on the (111), (211), (510), and (110) planes are presented in figs. 1, 2, 4 and 5, respectively. The data set collected from the (100) plane is presented in fig. 3 as plots of the logarithm of the total energy distribution. The enhancement factors have not been calculated for this plane because of the difficulty in determining a meaningful low energy slope for the free electron "background". All the curves are plotted against energy relative to E v + h ~ , where E F is taken as the average value for all of the data. Each curve is labelled with the appropriate photon energy. A photon energy of zero corresponds to field emission. A change in slope in the enhancement factor is considered to be a "feature" which departs from free electron theory and which may be compared from curve to curve. The energy at which the feature occurs is found by extrapolating linear portions of the enhancement factor, as is indicated in the plots by the dashed lines. A comparison of the plots shows that on each plane the enhancement factors for different photon energies are qualitatively very similar. Some of the curves for a photon energy of 3.536 eV are an exception to this general observation. This photon energy is large enough that the high energy electrons in these distributions are emitted above the peak of the surface potential barrier, and therefore the transmission probability does not depend exponentially on energy. Furthermore, if the photocurrent passes above the peak of the barrier, it will include contributions from states with larger transverse momentum than when lower photon energies are used. Thus the enhancement factors for 3.536 eV photons are expected to be different than those at lower photon energy due to a qualitative difference in the surface potential barrier at the final state energy. This dependence of the distribution on the form of the barrier demonstrates that photofield emission can be used as a sensitive probe of the surface potential.
D. Venus, M.J.G. Lee / P E S of surface electronic states of W
486
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Fig. 1. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (111) plane of tungsten when p-polarized light is incident at 90 °. The appropriate photon energies label each curve. The curves have been displaced vertically for clarity. Vertical lines mark the mean energy of qualitatively similar features. The numbering of the lines refers to table 1.
The energies of qualitatively similar features within a data set are compared in table 1. Vertical lines are drawn on the plots (figs. 1-5) at the mean energy relative to Ev + h ~ for the occurrence of each feature. Each line is labelled with a number which refers to the corresponding entry in table 1. The standard deviation, o, about the mean energy at which a feature occurs in a data set is between 10 and 35 mV. Since this standard deviation is comparable to the uncertainty in determining the absolute Fermi energy ( = 17 mV) and to the channel width with which the data are recorded (24 mV), the observed deviations are not significant. Thus all the traces from a given plane show a common enhancement factor shifted to higher energy by the photon energy hw. Minor variations are likely the result of the dependence of the transition matrix elements on the photon energy.
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Comparison of the photofield emission data with the field emission distribution (hto = 0) and with earlier published field emission data establishes that the common shifted enhancement factor is indeed that observed in field emission. Many features of the kind observed in the photofield emission data may be identified in the field emission data of Plummer and Bell [9] (which are reproduced in fig. 6) and in the tabulated data of Plummer and Gadzuk [10]. The energies at which these features occur are also entered in table 1. Since Plummet and Gadzuk report only the energies of the peaks of the enhancement factors, some features have no entry in the last column of the table. Features indicated by " S / N " are missing from the photofield emission data but would occur at low energies where the signal is masked by the tail of the
D. Venus. M.J.G. Lee / PES of surface electronic states of W
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ENERGY RELATIVE TO EF+hW(eV) Fig. 3. N a t u r a l l o g a r i t h m o f t h e p h o t o f i e l d e m i s s i o n d i s t r i b u t i o n s f r o m t h e (100) p l a n e o f t u n g s t e n w h e n p - p o l a r i z e d light is i n c i d e n t at 90 ° . T h e p l o t t i n g c o n v e n t i o n s a r e d e f i n e d in t h e c a p t i o n to fig. 1.
field emission distribution. Features indicated by " A " are absent from the photofield emission data even though they would occur at an energy where noise is not a problem. Features indicated by " N / A " occur in the photofield emission data, but no measurements for the appropriate geometry are reported by Plummer and Bell or Plummer and Gadzuk. Reference to table 1 shows that for emission from the (111) and (211) planes there is a one-to-one correspondence between the features in the present and earlier data, as well as good agreement as to the energies at which these features occur. The two sets of previously published results differ from each other by about as much as either differs from the present data. The differences are not systematic. The photofield emission data from the (100) plane show the pronounced peak at - 0 . 3 5 eV observed in the earlier field
D. Venus, M.J.G. Lee / PES o/surface electronic states of W I
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Fig. 4. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (510) plane of tungsten when p-polarized light is incident at 90 ° . The plotting conventions are defined in the caption to fig. 1.
emission data and in the photofield emission data of Gao and Reifenberger [11], but not the two. peaks observed at lower energy in field emission. The lowest peak may be masked by noise, but it is not immediately obvious why the peak near - 0 . 7 8 eV is not observed. Surface contamination is not responsible, since Plummet and Gadzuk comment that this peak is much less sensitive to a small coverage of adsorbed gas than the peak at - 0 . 3 5 eV. A close comparison of the data from the (510) plane is not possible since Plummer and Bell do not report results for this plane, while Plummer and Gadzuk report results only for the nearby (310) plane. However, the data are consistent with a small dispersion of the high energy peak between the (510) and (310) planes. The peak near - 1.45 eV is not seen because of noise. The
490
D. Venus, M.J.G. Lee / PES of surface electronic states of W I
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ENERGY RELATIVE TO E v + h w ( e V ) Fig. 5. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (110) plane of tungsten when p-polarized light is incident at 90 °. The plotting conventions are defined in the caption to fig. 1.
enhancement factors for the (110) photofield emission data are very flat, in agreement with previous results. The count rate was large enough to investigate the energy range E v + h0~ < - 0 . 5 eV only when a photon energy of 3.536 eV was used. In this enhancement factor a series of weak features seen in the previously published field emission data are reproduced. There is satisfactory agreement as to the energies at which these features occur. Discrepancies are at least in part due to the difficulty in extrapolating the linear portions of the curve when the features are weak. Features at lower energy are masked by noise. Data for lower photon energies are not reported for the (110) plane because of the low count rate. The origin of the peaks in the field emission enhancement factors has been discussed by Plummer [24]. Most are thought to be due to high surface density
491
D. Venus, M.J.G. Lee / PES of surface electronic states of W
Table 1 Summary of features in the p-polarized data sets Plane
Feature
No.
(E - E F - h to) _ ~ (eV)
Plummer and Bell [9] (eV)
(111)
1 2 3 4
6 5 1 1
-0.688_+0.016 -0.925 +0.031 - 1.30 -1.56
-0.66 -0.92 - 1.34 -1.59
-0.70
1 2 3 4 5 6
5 5 5 1 1
-0.338+0.014 - 0.635 + 0.021 - 0.794 + 0.034 - 1.20 - 1.44 S/N
-0.36 - 0.66 - 0.75 - 1.13 - 1.40 - 1.58
-0.37 - 0.69
1 2 3 4 5
7 7 7
-0.234+0.013 -0.358+0.005 - 0.563 + 0.013 A A
-0.34 - 0.54 - 0.74 -1.23
-0.37
1 2 3
4 4
-0.354+0.022 -0.711 +0.030 A
N/A N/A N/A
-0.39 a)
1 2 3 4 5
1 1 1 1
-0.47 -0.58 - 0.79 -0.89 S/N
-0.40 -0.53 - 0.73 -0.91 - 1.17
= 0.5
(211)
(100)
(510)
(110)
"Plummer and Gadzuk [10] (eV)
- 1.33
- 1.50
- 0.78 -1.50
- 1.45 ~)
a) Measured from (310) plane.
of states regions in the surface band structure close to the origin of the surface B r i l l o u i n z o n e . T o a f i r s t a p p r o x i m a t i o n , t h e s u r f a c e b a n d s t r u c t u r e is g i v e n b y p r o j e c t i n g t h e b u l k b a n d s t r u c t u r e o n t o t h e s u r f a c e B r i l l o u i n z o n e [25]. T h u s the peaks may equivalently be associated with high density of states regions in the bulk band structure along the surface normal in the extended zone scheme. Features near -0.7 eV in tungsten correspond to the bottom of bands which t e r m i n a t e a t a F7+ p o i n t a t t h e z o n e c e n t r e , a n d f e a t u r e s n e a r - 1 . 3 e V c o r r e s p o n d t o t h e t o p o f b a n d s w h i c h t e r m i n a t e a t a F8+ p o i n t . B e t w e e n t h e s e e n e r g i e s t h e r e is a g a p i n t h e a l l o w e d s t a t e s a t F. T h i s g a p is a b s o l u t e a l o n g t h e ( 1 1 1 ) a n d ( 2 1 1 ) d i r e c t i o n s , b u t is c r o s s e d b y o t h e r b a n d s a t l a r g e r K z i n t h e other directions. The enhancement factors have a dip between these energies corresponding to the low density of states region in the band gap. The v a r i a t i o n i n t h e e n e r g i e s o f t h e f e a t u r e s f r o m p l a n e t o p l a n e is a c o n s e q u e n c e of the details of how the bands approach F in the different directions, and,
D. Venus, M.J.G. Lee / PES of surface electronic states of W
492
I
5.0 o
I
~
I
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Z
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.'.
.
_
-"
.,
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i
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I
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0
ENERGY RELATIVE TO EF (eV) Fig. 6. Enhancement factors (expressed as the natural logarithm) for field emission from tungsten as reported by Plummer and Bell. The curves are labelled by the appropriate plane indices and are displaced for clarity. The experimental data are reproduced from ref. [9], and the construction lines are added. Numbers which label features on the curves refer to entries in table 1.
perhaps more importantly, of how the wavefunction amplitude at the surface, I CZp i 2, weights the density of states. The main feature which cannot be explained by projecting the bulk band structure onto the surface is the Swanson hump at - 0 . 3 5 eV in the emission from the (100) plane. This peak in the emission from the (100) plane of tungsten was originally identified as a surface feature by observing that its amplitude in field emission is very sensitive to the adsorption of gases [26]. Subsequent experiments have determined the symmetry and the dispersion of this and other peaks in the surface Brillouin zone using photoemission [27,28] techniques, and have established that there are three separate bands of surface resonance states in the surface Brillouin zone normal to the <100) axis of tungsten. Two of these three bands are close enough to E v at small transverse wavevector that they may be seen in field emission distributions. The band higher in energy causes the Swanson hump. This band has even symmetry and exists even in the limit of zero transverse wavevector. The second band is observed as the peak at - 0 . 7 8 eV. This band has odd symmetry and does not exist in the limit of zero transverse wavevector. Consequently, the field
D. Venus, M.J.G. Lee / PES of surface electronic states of W
493
emission current from this state produces a peak much smaller than the Swanson hump. Recent calculations of the two-dimensional band structure of the (100) plane of tungsten predict surface resonance bands whose dispersion is in good agreement with the experimental results [25,29]. The symmetry groups of these surface resonance states determine whether or not they can couple to the perturbing dipole electromagnetic field. Hermanson [30] has shown that for normal photoemission from the (100) plane of tungsten, only the normal component of the vector potential can couple to states of even symmetry and only the parallel component of the vector potential can couple to states of odd symmetry. In photofield emission, the effect of the surface barrier is to restrict emission to states with small transverse wavevector. These selection rules should therefore be only slightly weakened. The p-polarized data with light at grazing incidence were collected with no component of A parallel to the surface. Thus photofield emission is dipole allowed from the higher energy resonance and dipole forbidden from the surface resonance at - 0 . 7 8 eV. This may explain why the lower peak is not observed in the photofield emission distributions. Since the (510) and (100) directions are separated by only 11 °, the broad peak in the enhancement factor for the (510) plane at - 0 . 3 5 eV is likely due to the same surface resonance. The breadth of the peak suggests that the surface resonance is not as well localized on the lower symmetry plane. Plummer [24] speculates that the peak in the data from the (211) plane and the shoulder in the data from the (110) plane near this energy are due to similar, but less pronounced, surface effects. In summary, the enhancement factors measured in photofield emission have the same form as those measured in field emission, but are shifted to higher energy by the photon energy h~0. These results, combined with the results of earlier polarization dependence experiments, are definitive evidence that the surface photoelectric effect is the predominant photoexcitation mechanism in photofield emission, so long as the electromagnetic vector potential has an appreciable component normal to the emitting facet. Photofield emission and field emission appear to be equally sensitive probes of the surface density of states below the Fermi energy. The experimental demonstration of the surface sensitivity of photofield emission using p-polarized light at grazing incidence supports the suggestion that it is a useful technique for the study of other surface properties such as adsorbate systems, nonlocal electromagnetic response at the surface, and the surface potential barrier. A notable feature of the present data is the lack of structure due to the surface density of states at the final state energy. This implies that the final density of states is free-electron-like and varies slowly with energy. This is surprising since the eigenstates of the crystal potential of tungsten between the Fermi level and the vacuum level are d-band states, and are certainly not free-electron-like. In fact, if the surface photoelectric effect is extinguished by
494
D. Venus, M.J.G. Lee / PES of surface electronic states of W
using normally incident light, structure due to the variation of the density of states of the d-bands in this same energy range is observed in the enhancement factors [31]. This suggests that the final states for surface excitation are the tunnelling states in the vacuum and that the strongest contribution to the surface transition matrix elements comes from the region just outside the metal, where the final density of states is only weakly influenced by the electronic states of the metal. It is not clear from the present experiments whether or not photofield emission can yield information about the final surface density of states of the crystal. It may be that unoccupied surface states and surface resonances which extend into the vacuum can be observed by the present technique (as they can in ultraviolet bremsstrahlung spectroscopy) but that none exist on the particular planes of tungsten and in the energy range studied here. Further experiments with other transition metals and with adsorbed overlayers are required to address this question.
4. Conclusions Photofield emission total energy distributions have been measured from the (100), (110), (111), (211), and (510) crystallographic planes of a tungsten field emitter illuminated by p-polarized light at grazing incidence at seven photon energies between 1.916 and 3.536 eV. Except at the highest photon energy (3.536 eV), the enhancement factors calculated from those distributions have the same form as those previously calculated from field emission distributions, but they are shifted to higher energy by one photon energy. Photofield emission distributions measured at the highest photon energy show departures from the field emission distributions because the photocurrent is emitted above the peak of the surface potential barrier. No evidence has been found for structure due to the surface density of states at the final state energy. These results indicate that the predominant mechanism of excitation by p-polarized light at grazing incidence is surface photoexcitation from occupied crystal states near the surface to free-electron-like tunnelling states in the vacuum. Further experiments are required to determine whether peaks in the final surface density of states due to surface resonances or antibonding states of adsorbed species can be observed through the surface photoelectric effect in photofield emission.
Acknowledgements We are grateful to Paul Donders for many discussions and for his help in solving numerous problems. The excellent technical assistance of K. Weisser of the Scarborough Academic Workshop and J. Shuve of the Electronics
D. Venus, M.J.G. Lee / PES of surface electronic states of W
495
Workshop are much appreciated. This work was supported in part by operating and equipment grants from the Natural Science and Engineering Research Council (NSERC) of Canada. One of us (DV) wishes to acknowledge financial support from NSERC and from the University of Toronto.
References [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]
I. Adawi, Phys. Rev. 134 (1963) A788. M.J.G. Lee, Phys. Rev. Letters 30 (1973) 1193. R. Reifenberger, H.A. Goldberg and M.J.G. Lee, Surface Sci. 83 (1979) 599. D. Havvig and R. Reifenberger, Surface Sci. 151 (1985) 128. A. Bagchi, Phys. Rev. B10 (1974) 542. C. Schwartz and M.W. Cole, Surface Sci. 115 (1982) 290. Y. Teisseyre, R. Haug and R. Coelho, Surface Sci. 87 (1979) 549. D. Venus and M..I.G. Lee, Surface Sci. 125 (1982) 452. E.W. Plummer and A.E. Bell, J. Vacuum Sci. Technol. 9 (1972) 583. E.W. Plummer and J.W. Gadzuk, Phys. Rev. Letters 25 (1970) 1439. Y. Gao and R. Reifenberger, unpublished. L.W. Swanson and L.C. Crouser, Phys. Rev. Letters 15 (1966) 389; Phys. Rev. Letters 19 (1967) 1179. E.L. Murphy and R.H. Good, Jr., Phys. Rev. 102 (1956) 1464. N. Nicolaou and A. Modinos, Surface Sci. 60 (1976) 527; Phys. Rev. B13 (1976) 1536; Phys. Rev. Bll (1975) 3687. W.L. Schaich and N.W. Ashcroft, Phys. Rev. B3 (1971) 2452. D. Venus, Photofield emission spectroscopy of tungsten, University of Toronto, PhD Thesis, 1985, unpublished. K.L. Kliewer, Surface Sci. 101 (1980) 57; Phys. Rev. B14 (1976) 1412. P.J. Feibelman, Phys. Rev. B12 (1975) 1319. N.D. Lang and W. Kohn, Phys. Rev. B1 (1970) 4555. P.J. Feibelman, Progr. Surface Sci. 12 (1983) 287. H.J. Levinson and E.W. Plummer, Phys. Rev. B24 (1981) 628. R.D. Young, Phys. Rev. 113 (1958) 110. D. Venus and M.J.G. Lee, Rev. Sci. Instr. 56 (1985) 1206. E.W. Plummer, in: Interactions on Metal Surfaces, Ed. R. Gomer (Springer, New York, 1975). L.F. Mattheiss and D.R. Hamann, Phys. Rev. B29 (1984) 5372. B.J. Waclawski and E.W. Plummet, Phys. Rev. Letters 29 (1972) 783. S.-L. Weng, E.W. Plummet and T. Gustafsson, Phys. Rev. B18 (1978) 1718. M.J. Holmes and T. Gustafsson, Phys. Rev. Letters 47 (1981) 443. S. Ohmishi, A.J. Freeman and E. Wimmer, Phys. B24 (1984) 5267. J. Hermanson, Solid State Commun. 22 (1977) 9. D. Venus and M.J.G. Lee, unpublished.
PHOTOFIELD EMISSION SPECTROSCOPY ELECTRONIC STATES OF TUNGSTEN
477
OF SURFACE
David VENUS Department of Physics, University of Toronto, Toronto, Ontario M5S IA 7, Canada
and M a r t i n J.G. L E E Department of Physics and Scarborough College, University of Toronto, Toronto, Ontario M5S IA 7, Canada
Received 29 July 1985; accepted for publication 29 January 1986 A novel photofield emission spectrometer has been used to observe departures from free electron behaviour in the total energy distributions of photofield emission from the (100), (110), (111), (211), and (510) facets of a tungsten field emitter. Measurements with p-polarized light at grazing incidence at seven photon energies in the visible and near ultraviolet show departures from free electron behaviour having the same form as those observed in field emission distributions, but shifted to higher energy by one photon energy. The shapes and polarization dependence of the distributions are precisely those expected for surface photoexcitation from electron states near the surface to vacuum tunnelling final states. These observations provide strong evidence that the surface photoeffect is the dominant mechanism of excitation by p-polarized light in photofield emission from tungsten.
I. Introduction P h o t o f i e l d emission is the p h e n o m e n o n w h e r e b y p h o t o e x c i t e d electrons escape f r o m a solid b y tunnelling to the v a c u u m in the presence of a large electric field a p p l i e d to the surface. It has features in c o m m o n with b o t h p h o t o e m i s s i o n a n d field emission. W h e r e a s field emission p r o b e s electronic states b e l o w the F e r m i level b y using large surface electric fields to allow tunnelling, a n d p h o t o e m i s s i o n p r o b e s u n b o u n d electronic states a b o v e the v a c u u m level b y using p h o t o e x c i t a t i o n , p h o t o f i e l d emission explores the i n t e r m e d i a t e energy range b e t w e e n the F e r m i level a n d the v a c u u m level. Electronic excitation in p h o t o f i e l d emission has been a t t r i b u t e d to the surface p h o t o e l e c t r i c effect [1]. This i n t e r p r e t a t i o n has been inferred f r o m the field d e p e n d e n c e o f the total p h o t o c u r r e n t [2], a n d b y fitting [3,4] the d i s t r i b u tions to free electron m o d e l s of p h o t o f i e l d emission which a s s u m e surface p h o t o e x c i t a t i o n [5,6]. M e a s u r e m e n t s of the p o l a r i z a t i o n d e p e n d e n c e of the p h o t o c u r r e n t [7,8] have also i n d i c a t e d that the d o m i n a n t m e c h a n i s m o f 0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
478
D, Venus, M.J.G. Lee / PES of surface electronic states of W
excitation is the surface photoelectric effect. This has led to the expectation that photofield emission is a very sensitive probe of electronic states at the surface of a metal, and that it may be useful in studies of the surface potential barrier, the detection of unoccupied surface states between the Fermi level and the vacuum level, the investigation of the nonlocal electromagnetic response of a metal surface, and studies of adsorbate systems. The dependence of the total energy distribution on the photon energy can serve as a further test of the hypothesis that surface excitation is the dominant photoexcitation mechanism in photofield emission. Field emission experiments have revealed significant departures from free electron behaviour in the surface density of states at the initial state energy [9,10]. The surface photoexcitation hypothesis implies that these same departures will also be observed in photofield emission distributions, but shifted to higher energy by the energy of one photon. The identification of the mechanism of excitation in photofield emission will not be conclusive until either such structure is observed or its absence is accounted for. Recent experimental findings of Gao and Reifenberger [11] have gone some way to addressing this question. They have measured the photofield emission total energy distribution from the (100) plane of tungsten with light of wavelength 477 nm, and report a large secondary peak in the distribution. This peak is attributed to surface photoexcitation from the well-known surface resonance (the "Swanson hump") on this plane 0.35 eV below the Fermi energy [12]. However, as their published data are restricted to a single photon energy, the dependence of the distribution on photon energy cannot be evaluated. In the present paper, photo field emission distributions are reported on the (100), (110), (111), (211), and (510) crystallographic planes of a tungsten field emitter, using p-polarized light at grazing incidence at seven photon energies ranging from 1.916 to 3.536 eV. The distributions measured on the (100) plane are consistent with that of Gao and Reifenberger, and many departures from free electron behaviour on other planes are reported for the first time in photofield emission. At each photon energy, departures from free electron behaviour in the photofield emission distributions are found to be in one-to-one correspondence with the initial state structure observed in field emission, providing conclusive evidence of surface excitation under these polarization conditions. This paper is divided into four sections. In section 2 the theory of surface excitation is reviewed. In section 3 the experimental procedure and results are described. Finally, the conclusions of this work are summarized in section 4. 2. Theory When a strong static electric field is applied to the surface of a metal, the surface potential changes from a step to a roughly triangular barrier, through
D. Venus, M.J.G. Lee / PES of surface electronicstates of W
479
which electrons may tunnel to the vacuum region. To observe the tunnelling of conduction electrons through the surface barrier ("field emission") from tungsten, an electric field of the order of 0.2 V/,~ is required. In order to obtain such a large field, the sample is chosen to be a thin wire, the end of which has been electrochemically etched to form a cone with a hemispherical endcap of radius --- 0.1/xm, called the "tip". Because of its small size the tip is a single crystal, with facets approximately 500 ,~ square normal to the low index crystallographic directions. The sample is spot welded to a tungsten support loop, and mounted in an ultra-high vacuum chamber operating in the 10-11 Torr range. A potential difference of = 2000 V applied between the tip and a conducting fluorescent screen creates the required field at the tip surface. The field emission current causes fluorescence on the screen, and a characteristic pattern is produced by the variations in the work function from one tip facet to another. The appearance of the field emission pattern indicates the state of cleanliness of the sample. The sample is cleaned whenever necessary by electrically heating the support loop to incandescence. Field emission from a single crystallographic direction may be selected by locating a small probe hole in the screen. The probe hole current can be analysed using appropriate electron optics to yield the total energy distribution of the electrons. In photofield emission, laser light with a photon energy less than the work function of the metal is focussed on the tip, and the total energy distribution of the photofield emission is measured. The Hamiltonian which describes the system of electrons is H o = - (h2/2m)V
(1)
2 + V(r) + H',
where V ( r ) = lie(,'),
Z < 0,
V(r)=V~(z)=-eFz-e2/az,
z>O.
The crystal potential Vc(r ) is the three-dimensional periodic potential inside a metal. The surface potential Vs(z ) has terms resulting from the applied electric field F and the image charge, and varies only in the direction of the surface normal, L The perturbing electromagnetic dipole interaction which causes photoexcitation is given by H ' = -ihe2m____c[ V . A ( r )
+A(r).
V] e -i*'t,
(2)
where - i h ~ 7 is the electron momentum and the monochromatic electromagnetic vector potential A (r)exp(-i~0t) is written in the gauge where the scalar potential is identically zero. Inside the metal, the eigenstates [M>, ( M stands for a set of quantum numbers) are solutions of the SchriSdinger equation for the crystal potential. In the vacuum region, the eigenstates are a product of free electron states of
D. Venus, M.J.G. Lee / PES of surface electronic states of W
480
transverse wavevector p parallel to the interface and complicated tunnelling functions in the z-direction. The tunnelhng function of energy E M, transverse momentum p, and unit amplitude at the surface, is well approximated far from the surface by a plane wave whose amplitude is equal to the tunnelling coefficient T(W~). Since T(W~) is an exponential function [13] of the normal energy
W~, = EM -- ( h2/2,, ) p 2, the majority of the emission transverse wavevector. Following the field emission the photoemission calculation emission current excited from written as [16] 2~re
JL = - - ~ - - E S ( E L + h ° ° - - E M )
(3) current will derive from states with small calculation of Nicolaou and Modinos [14] and of Schaich and Ashcroft [15], the photofield the initially unperturbed state L ) may be
IDMLI 2 E I C ~ l 2 IT(WMp) I 2
M
(4)
p
In this expression the optical transition matrix element DML is given by
DML =
ihe 2mc(MIV.A(r)+A(r).VIL),
(5)
and the delta function allows transitions to final states I M ) only if the transition conserves energy. The C~p are the expansion coefficients required at the interface (z = 0) to match the crystal state of energy E M and transverse momentum k M to the tunnelling eigenfunction of energy E M and transverse momentum p which propagates in the the + z direction. The sum includes all those transverse momenta which satisfy the condition
P:kM+g,
(6)
where g is a surface reciprocal lattice vector. This matching condition implies that transmission through a plane surface conserves the transverse crystal momentum only to within a surface reciprocal lattice vector. The total energy distribution of the photofield emission current from all crystal states is
J ( E ) = ~'~f(EL) JL [ 1 - - f ( E M ) ] 8 ( E - E M ) , L
(7)
where f ( E ) is the Fermi-Dirac distribution function. The matrix element DML describes the photoexcitation step in photofield emission. It is instructive to rewrite the matrix element in the form
DML = f l ( M I [ V . A ( r ) ]
[L) + 2 f l ( M l A ( r ) . ~7IL),
(8)
where fl = - i h e / 2 m c . The first term in this equation represents transitions induced by a spatial variation of the vector potential. If the incident light has a
D. Venus, M.J.G. Lee / PES of surface electronic states of W
481
component of vector potential normal to the plane metal-vacuum interface, the vector potential will vary rapidly in the interface region and cause this term to contribute to surface photoexcitation. In a simple local response theory, this rapid variation takes the form of a discontinuity in the component of the vector potential normal to the interface, while the component of the vector potential parallel to the interface is conserved and does not excite electronic transitions at the surface. This contribution to the surface excitation is essentially unchanged if a more physically correct nonlocal response theory for the surface is considered [17]. An additional effect is incorporated in the nonlocal calculation of Feibelman [18], who uses the self-consistent surface potential calculated by Lang and Kohn [19]. In this potential, the metal surface is defined by the sharp boundary of the positively charged background "jellium". The electron density is no longer confined to the z < 0 side of the interface, but spills out into the vacuum region very close to the surface in a sigmoid distribution. Feibelman demonstrates that at frequencies below the bulk plasma frequency, the normal component of the vector potential has a large peak in the surface region which greatly "overshoots" the change expected from the local theory. This peak decays to the local theory value via Friedel-like oscillations. The peak may be understood qualitatively [20] as a plasma oscillation localized within the region of low electron density at z > 0. Photoemission experiments by Levinson and Plummer [21] have verified that the surface peak in A z due to this effect makes a significant contribution to the surface photoexcitation for aluminum. This effect is also expected to be important in photofield emission, since the optical frequencies used in photofield emission are typically much smaller than the plasma frequency. The second term in the matrix element in eq. (8) can also make a contribution to surface photoexcitation, but through a slightly different mechanism. Even if the vector potential has zero divergence, this term will be non-zero. Under these circumstances, it may be evaluated using the commutator identity [1]
2flA • ~ M I W I L) =- ~ A .
~MIITV(r) I L).
(9)
In the surface region the potential V(r) is given by the surface potential V~(z), and this term results in the surface photoelectric effect in its original, and narrowest, definition. Since the surface potential varies only in the normal direction, only the normal component of the vector potential is effective in exciting these transitions. Thus the surface excitation described by both terms in eq. (8) will be maximized by choosing the polarization vector of the incident light to be normal to the surface. This may be accomplished by using p-polarized light at grazing incidence. Under these conditions, previous studies [8] have shown that approximately 0.5% of the total photofield emission
D. Venus, M.J.G. Lee / PES of surface electronic states" of W
482
current has a polarization dependence that is inconsistent with surface excitation. Field emission and photofield emission are sensitive to surface electronic states because the matching coefficients C~p in eq. (4) weight the emission current approximately as the surface density of states at the final state energy. In photofield emission due to surface excitation, the photocurrent further depends on the surface electronic states because the matrix elements DML are appreciable only close to the surface. Since surface optical transitions need not conserve the component of electron m o m e n t u m normal to the interface, and since only states with small transverse wavevector contribute significantly to the emitted current, the matrix elements weight the current approximately as the product of the initial and final surface densities of states at k = 0. Together, these effects are expected to produce departures from free electron behaviour in the total energy distributions. The transitions will involve the same initial states as in field emission. Therefore, those departures from free electron behaviour observed in field emission energy distributions [9,10] are expected to appear also in the photofield emission distributions, but shifted to higher energy by the energy of one photon. It is not clear, a priori, whether the final density of states will be governed principally by the distribution of electron states at the surface of the crystal or in the vacuum tunnelling region. Experimental studies of the total energy distribution may help to resolve this question. In order to identify departures from free electron behaviour due to surface photoexcitation, it is useful to reduce eq. (4) to the limit of the free electron model. The potential in a free electron metal is not periodic, so the sum over p collapses to the single term p = k M. Since the crystal states are free electron states, C~te is unity, and the sums over L and M represent integrals over total wavevector states K L and K M. The matrix elements involve only functions of the spatial coordinate z, and may be written as
I DML 12 =
(47r2/ct)
lAMa [2 3(kM-- kL),
(10)
where AML is the matrix d e m e n t between the one-dimensional wavefunctions in the z-direction, and ct is the area of the interface. These integrals may be transformed to integrals over total energy, normal energy, and azimuthal angle, and evaluated to yield an expression for the total energy distribution at the detector,
J(E) = 4m2e122/(Ir3hSct) f ( E - h~) xf E
[1 - f ( E ) ]
-1/2 laMLt 2 Ir(W)l
(11)
" Vo+hw
in agreement with the results of Schwartz and Cole [6]. In this equation, 12 is the volume within which the wavefunctions are normalized, and V0 is the
D. Venus, M.J.G. Lee / PES of surface electronic states of W
483
energy at the bottom of the free electron conduction band. The barrier tunnelling probability may be evaluated in the WKB approximation and expanded about E F + h~ as [13] IT(W) 12 = e -c e t w - e ~ - h ~ ' ) / a ,
(12)
where c = 4 ( 2 m ~ , * ) a / 2 v ( y ) , d = h e F / [ 2 ( 2 m e p * ) a / 2 t ( y ) ] , y = ( e 3 F ) l / 2 / e p *, ~* = ]~-h~0 I, ~ is the work function and v ( y ) and t ( y ) are elliptic integrals defined in ref. [13]. Since the factor [ ( W - V o ) ( W - h c o - V 0 ) ] -1/2 varies slowly compared to the exponential, it may be approximated by its endpoint. If the matrix element is assumed to be a constant, ]A[ 2, for all initial and final states separated by one photon energy, then eq. (11) yields 8mf22 [ 1 - f ( E ) ]
J(E)-
[AI2 [ ( E - V o ) ( E - h ~ -
Vo)] -'/2 J~-E(E),
•/.r h 2o~
(13a) where J~E(E)
reed
-
-
.
2--~3/(e-
h~) IT(E)
[2
(13b)
reduces to the field emission current density [22] evaluated with an effective Fermi level E F + h w. As the factor [ ( E - V o ) ( E - h w - ~)]-1/2 is slowly varying in comparison with J~E(E), the total energy distribution in photofield emission from a free electron metal is given to a good approximation by shifting the field emission distribution by the photon energy h0~. A plot of the logarithm of the current against energy is roughly triangular in shape. The constant positive slope at low energy is 1 / d , and depends only on the barrier transmission. The constant negative slope at high energy is 1 / d - 1 / k T , and results from the barrier transmission and the cutoff of the Fermi occupation factor. Extrapolations of these constant slope regions intersect at E F + hw. Departures from free electron behaviour will therefore appear as changes in slope on the linear portions of the logarithmic plot.
3. Experiment and results The photofield emission spectrometer used to measure the total energy distributions has been described elsewhere [23]. This apparatus has two important features that were not available in the apparatus used in previous studies. The first of these features is a sample positioner which allows the laser light to fall on any crystallographic facet of the field emitter with any desired angles of incidence and linear polarization. If the facet is illuminated by p-polarized light at grazing incidence, the photocurrent due to surface excita-
484
D. Venus, M.J.G. Lee / PES of surface electronic states of W
tion is expected to dominate the total photocurrent. The second feature is a deflection energy analyser consisting of two 127 ° cylindrical analysers in tandem. This analyser has a resolution function with a full width at half m a x i m u m height of (0.0390 + 0.0006)E 0, where E 0 is the pass kinetic energy of the analyser, and a peak signal-to-noise ratio of 2 × 10 4. The large signalto-noise ratio makes it possible to extend measurements of the total energy distribution over four orders of magnitude. By plotting the logarithm of the distribution against energy, it is possible to detect departures from free electron behaviour. Total energy distributions have been measured for five well defined regions of the tungsten field emission pattern, corresponding to the (100), (110), (111), (211) and (510) crystallographic directions. The light source was a krypton ion laser which operates at seven wavelengths in the visible and near ultraviolet corresponding to photon energies between 1.916 and 3.536 eV. Low count rates from planes with large work functions, such as the (110), made measurements at the lower photon energies very difficult. These data are not included since they have too much shot noise to be of use. Field emission and photofield emission distributions were recorded by means of a computer controlled data acquisition system. The pass energy of the electron energy analyser was set to 2 eV, giving an instrumental function with a full width at half maximum height of approximately 80 mV. The distribution was divided into about one hundred equal intervals, or channels, each having a width of approximately 24 mV. The entire distribution was collected every 200 ms by sweeping through the channels sequentially and recording the counts accumulated in each channel in 2 ms. The results from each sweep were added to average out the effects of alignment drift and slow tip contamination. Data were accumulated for between 15 and 30 min, depending on the size of the statistical noise. A F o w l e r - N o r d h e i m plot was taken to establish the field constant of the tip for each facet. TypicallY,othe field at which the distributions were measured was within 10% of 0.25 V / A . In order to resolve departures from free electron behaviour in the total energy distribution, it is necessary to remove the influence of the barrier transmission probability by subtracting the free electron "background". According to eq. (13), the free electron contribution to the photofield emission distribution may be parameterized as In J ( E ) = l n In J ( E )
A-(E-E
v-h~o)m
1,
= In A - ( E - E F - h r o ) m 2,
-5kT,
(14a)
E - E v - ho~ > 5 k T ,
(14b)
E-E
v-boo<
where A is the amplitude of the distribution and the slopes rn~ and m 2 may be related to the temperature and applied field using eq. (13). If straight lines are fitted to the linear portion of the logarithm of the distribution at high and low energy, the slopes and intercepts of these lines determine the four parameters A , E v, m~ and m 2. Such fits have been made to the data measured on all five
D. Venus, M.J.G. Lee / PES of surface electronic states of W
485
planes, and yield a consistent value of the Fermi energy with an uncertainty of + 17 meV. To bring the departures from free electron behaviour into greater prominence, the enhancement factor R ( E ) has been calculated as the difference between the natural logarithms of the experimental and fitted free electron curves. R(E)
= In J e x p ( E ) - In Jfree(E).
(15)
The energy range in which the enhancement factor contains useful information is restricted to the linear portion of the low energy side of the fitted distribution. The linear portion of the high energy side of the distribution is so steep that deviations from linearity cannot be reliably detected. The rounded portion at the apex of the fitted distribution is the energy range where the Fermi function varies most rapidly. Structure in this region of the enhancement factor is sensitive to small changes in the fit and is not reproducible. At very low energies, the enhancement factor becomes large because the field emission distribution overlaps the photofield emission distribution. Enhancement factors for the data sets measured on the (111), (211), (510), and (110) planes are presented in figs. 1, 2, 4 and 5, respectively. The data set collected from the (100) plane is presented in fig. 3 as plots of the logarithm of the total energy distribution. The enhancement factors have not been calculated for this plane because of the difficulty in determining a meaningful low energy slope for the free electron "background". All the curves are plotted against energy relative to E v + h ~ , where E F is taken as the average value for all of the data. Each curve is labelled with the appropriate photon energy. A photon energy of zero corresponds to field emission. A change in slope in the enhancement factor is considered to be a "feature" which departs from free electron theory and which may be compared from curve to curve. The energy at which the feature occurs is found by extrapolating linear portions of the enhancement factor, as is indicated in the plots by the dashed lines. A comparison of the plots shows that on each plane the enhancement factors for different photon energies are qualitatively very similar. Some of the curves for a photon energy of 3.536 eV are an exception to this general observation. This photon energy is large enough that the high energy electrons in these distributions are emitted above the peak of the surface potential barrier, and therefore the transmission probability does not depend exponentially on energy. Furthermore, if the photocurrent passes above the peak of the barrier, it will include contributions from states with larger transverse momentum than when lower photon energies are used. Thus the enhancement factors for 3.536 eV photons are expected to be different than those at lower photon energy due to a qualitative difference in the surface potential barrier at the final state energy. This dependence of the distribution on the form of the barrier demonstrates that photofield emission can be used as a sensitive probe of the surface potential.
D. Venus, M.J.G. Lee / P E S of surface electronic states of W
486
I
4.0
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I i
-0.8
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-0.5
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TO EF+h~o (eV)
Fig. 1. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (111) plane of tungsten when p-polarized light is incident at 90 °. The appropriate photon energies label each curve. The curves have been displaced vertically for clarity. Vertical lines mark the mean energy of qualitatively similar features. The numbering of the lines refers to table 1.
The energies of qualitatively similar features within a data set are compared in table 1. Vertical lines are drawn on the plots (figs. 1-5) at the mean energy relative to Ev + h ~ for the occurrence of each feature. Each line is labelled with a number which refers to the corresponding entry in table 1. The standard deviation, o, about the mean energy at which a feature occurs in a data set is between 10 and 35 mV. Since this standard deviation is comparable to the uncertainty in determining the absolute Fermi energy ( = 17 mV) and to the channel width with which the data are recorded (24 mV), the observed deviations are not significant. Thus all the traces from a given plane show a common enhancement factor shifted to higher energy by the photon energy hw. Minor variations are likely the result of the dependence of the transition matrix elements on the photon energy.
D. Venus, M.J.G. Lee / PES of surface electronic states of IV I
I
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W (211) P pol.
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487
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Fig. 2. E n h a n c e m e n t factors (expressed as the n a t u r a l l o g a r i t h m ) for p h o t o f i e l d e m i s s i o n from the (211) p l a n e of tungsten w h e n p-polarized light is i n c i d e n t at 90 ° . The p l o t t i n g c o n v e n t i o n s are d e f i n e d in the caption to fig. 1.
Comparison of the photofield emission data with the field emission distribution (hto = 0) and with earlier published field emission data establishes that the common shifted enhancement factor is indeed that observed in field emission. Many features of the kind observed in the photofield emission data may be identified in the field emission data of Plummer and Bell [9] (which are reproduced in fig. 6) and in the tabulated data of Plummer and Gadzuk [10]. The energies at which these features occur are also entered in table 1. Since Plummet and Gadzuk report only the energies of the peaks of the enhancement factors, some features have no entry in the last column of the table. Features indicated by " S / N " are missing from the photofield emission data but would occur at low energies where the signal is masked by the tail of the
D. Venus. M.J.G. Lee / PES of surface electronic states of W
488
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///
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0.0 I
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ENERGY RELATIVE TO EF+hW(eV) Fig. 3. N a t u r a l l o g a r i t h m o f t h e p h o t o f i e l d e m i s s i o n d i s t r i b u t i o n s f r o m t h e (100) p l a n e o f t u n g s t e n w h e n p - p o l a r i z e d light is i n c i d e n t at 90 ° . T h e p l o t t i n g c o n v e n t i o n s a r e d e f i n e d in t h e c a p t i o n to fig. 1.
field emission distribution. Features indicated by " A " are absent from the photofield emission data even though they would occur at an energy where noise is not a problem. Features indicated by " N / A " occur in the photofield emission data, but no measurements for the appropriate geometry are reported by Plummer and Bell or Plummer and Gadzuk. Reference to table 1 shows that for emission from the (111) and (211) planes there is a one-to-one correspondence between the features in the present and earlier data, as well as good agreement as to the energies at which these features occur. The two sets of previously published results differ from each other by about as much as either differs from the present data. The differences are not systematic. The photofield emission data from the (100) plane show the pronounced peak at - 0 . 3 5 eV observed in the earlier field
D. Venus, M.J.G. Lee / PES o/surface electronic states of W I
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489
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W (510) P pol. 3,536***
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,r
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I/L
-0.8
1
-0.5
ENERGY RELATIVE TO E F + h w
-0.2
(eV)
Fig. 4. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (510) plane of tungsten when p-polarized light is incident at 90 ° . The plotting conventions are defined in the caption to fig. 1.
emission data and in the photofield emission data of Gao and Reifenberger [11], but not the two. peaks observed at lower energy in field emission. The lowest peak may be masked by noise, but it is not immediately obvious why the peak near - 0 . 7 8 eV is not observed. Surface contamination is not responsible, since Plummet and Gadzuk comment that this peak is much less sensitive to a small coverage of adsorbed gas than the peak at - 0 . 3 5 eV. A close comparison of the data from the (510) plane is not possible since Plummer and Bell do not report results for this plane, while Plummer and Gadzuk report results only for the nearby (310) plane. However, the data are consistent with a small dispersion of the high energy peak between the (510) and (310) planes. The peak near - 1.45 eV is not seen because of noise. The
490
D. Venus, M.J.G. Lee / PES of surface electronic states of W I
I
I
W (llO) Ppol,
5.0
0.0
4.0 n-
Iz
3.oF
w O0
W
OI
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ENERGY RELATIVE TO E v + h w ( e V ) Fig. 5. Enhancement factors (expressed as the natural logarithm) for photofield emission from the (110) plane of tungsten when p-polarized light is incident at 90 °. The plotting conventions are defined in the caption to fig. 1.
enhancement factors for the (110) photofield emission data are very flat, in agreement with previous results. The count rate was large enough to investigate the energy range E v + h0~ < - 0 . 5 eV only when a photon energy of 3.536 eV was used. In this enhancement factor a series of weak features seen in the previously published field emission data are reproduced. There is satisfactory agreement as to the energies at which these features occur. Discrepancies are at least in part due to the difficulty in extrapolating the linear portions of the curve when the features are weak. Features at lower energy are masked by noise. Data for lower photon energies are not reported for the (110) plane because of the low count rate. The origin of the peaks in the field emission enhancement factors has been discussed by Plummer [24]. Most are thought to be due to high surface density
491
D. Venus, M.J.G. Lee / PES of surface electronic states of W
Table 1 Summary of features in the p-polarized data sets Plane
Feature
No.
(E - E F - h to) _ ~ (eV)
Plummer and Bell [9] (eV)
(111)
1 2 3 4
6 5 1 1
-0.688_+0.016 -0.925 +0.031 - 1.30 -1.56
-0.66 -0.92 - 1.34 -1.59
-0.70
1 2 3 4 5 6
5 5 5 1 1
-0.338+0.014 - 0.635 + 0.021 - 0.794 + 0.034 - 1.20 - 1.44 S/N
-0.36 - 0.66 - 0.75 - 1.13 - 1.40 - 1.58
-0.37 - 0.69
1 2 3 4 5
7 7 7
-0.234+0.013 -0.358+0.005 - 0.563 + 0.013 A A
-0.34 - 0.54 - 0.74 -1.23
-0.37
1 2 3
4 4
-0.354+0.022 -0.711 +0.030 A
N/A N/A N/A
-0.39 a)
1 2 3 4 5
1 1 1 1
-0.47 -0.58 - 0.79 -0.89 S/N
-0.40 -0.53 - 0.73 -0.91 - 1.17
= 0.5
(211)
(100)
(510)
(110)
"Plummer and Gadzuk [10] (eV)
- 1.33
- 1.50
- 0.78 -1.50
- 1.45 ~)
a) Measured from (310) plane.
of states regions in the surface band structure close to the origin of the surface B r i l l o u i n z o n e . T o a f i r s t a p p r o x i m a t i o n , t h e s u r f a c e b a n d s t r u c t u r e is g i v e n b y p r o j e c t i n g t h e b u l k b a n d s t r u c t u r e o n t o t h e s u r f a c e B r i l l o u i n z o n e [25]. T h u s the peaks may equivalently be associated with high density of states regions in the bulk band structure along the surface normal in the extended zone scheme. Features near -0.7 eV in tungsten correspond to the bottom of bands which t e r m i n a t e a t a F7+ p o i n t a t t h e z o n e c e n t r e , a n d f e a t u r e s n e a r - 1 . 3 e V c o r r e s p o n d t o t h e t o p o f b a n d s w h i c h t e r m i n a t e a t a F8+ p o i n t . B e t w e e n t h e s e e n e r g i e s t h e r e is a g a p i n t h e a l l o w e d s t a t e s a t F. T h i s g a p is a b s o l u t e a l o n g t h e ( 1 1 1 ) a n d ( 2 1 1 ) d i r e c t i o n s , b u t is c r o s s e d b y o t h e r b a n d s a t l a r g e r K z i n t h e other directions. The enhancement factors have a dip between these energies corresponding to the low density of states region in the band gap. The v a r i a t i o n i n t h e e n e r g i e s o f t h e f e a t u r e s f r o m p l a n e t o p l a n e is a c o n s e q u e n c e of the details of how the bands approach F in the different directions, and,
D. Venus, M.J.G. Lee / PES of surface electronic states of W
492
I
5.0 o
I
~
I
(It1)
4.0I,,21
Z
OiO)
u.l 1.00
-2.0
.'.
.
_
-"
.,
1
~ i
-t.5
i
-I.O
I
-0.5
0
ENERGY RELATIVE TO EF (eV) Fig. 6. Enhancement factors (expressed as the natural logarithm) for field emission from tungsten as reported by Plummer and Bell. The curves are labelled by the appropriate plane indices and are displaced for clarity. The experimental data are reproduced from ref. [9], and the construction lines are added. Numbers which label features on the curves refer to entries in table 1.
perhaps more importantly, of how the wavefunction amplitude at the surface, I CZp i 2, weights the density of states. The main feature which cannot be explained by projecting the bulk band structure onto the surface is the Swanson hump at - 0 . 3 5 eV in the emission from the (100) plane. This peak in the emission from the (100) plane of tungsten was originally identified as a surface feature by observing that its amplitude in field emission is very sensitive to the adsorption of gases [26]. Subsequent experiments have determined the symmetry and the dispersion of this and other peaks in the surface Brillouin zone using photoemission [27,28] techniques, and have established that there are three separate bands of surface resonance states in the surface Brillouin zone normal to the <100) axis of tungsten. Two of these three bands are close enough to E v at small transverse wavevector that they may be seen in field emission distributions. The band higher in energy causes the Swanson hump. This band has even symmetry and exists even in the limit of zero transverse wavevector. The second band is observed as the peak at - 0 . 7 8 eV. This band has odd symmetry and does not exist in the limit of zero transverse wavevector. Consequently, the field
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emission current from this state produces a peak much smaller than the Swanson hump. Recent calculations of the two-dimensional band structure of the (100) plane of tungsten predict surface resonance bands whose dispersion is in good agreement with the experimental results [25,29]. The symmetry groups of these surface resonance states determine whether or not they can couple to the perturbing dipole electromagnetic field. Hermanson [30] has shown that for normal photoemission from the (100) plane of tungsten, only the normal component of the vector potential can couple to states of even symmetry and only the parallel component of the vector potential can couple to states of odd symmetry. In photofield emission, the effect of the surface barrier is to restrict emission to states with small transverse wavevector. These selection rules should therefore be only slightly weakened. The p-polarized data with light at grazing incidence were collected with no component of A parallel to the surface. Thus photofield emission is dipole allowed from the higher energy resonance and dipole forbidden from the surface resonance at - 0 . 7 8 eV. This may explain why the lower peak is not observed in the photofield emission distributions. Since the (510) and (100) directions are separated by only 11 °, the broad peak in the enhancement factor for the (510) plane at - 0 . 3 5 eV is likely due to the same surface resonance. The breadth of the peak suggests that the surface resonance is not as well localized on the lower symmetry plane. Plummer [24] speculates that the peak in the data from the (211) plane and the shoulder in the data from the (110) plane near this energy are due to similar, but less pronounced, surface effects. In summary, the enhancement factors measured in photofield emission have the same form as those measured in field emission, but are shifted to higher energy by the photon energy h~0. These results, combined with the results of earlier polarization dependence experiments, are definitive evidence that the surface photoelectric effect is the predominant photoexcitation mechanism in photofield emission, so long as the electromagnetic vector potential has an appreciable component normal to the emitting facet. Photofield emission and field emission appear to be equally sensitive probes of the surface density of states below the Fermi energy. The experimental demonstration of the surface sensitivity of photofield emission using p-polarized light at grazing incidence supports the suggestion that it is a useful technique for the study of other surface properties such as adsorbate systems, nonlocal electromagnetic response at the surface, and the surface potential barrier. A notable feature of the present data is the lack of structure due to the surface density of states at the final state energy. This implies that the final density of states is free-electron-like and varies slowly with energy. This is surprising since the eigenstates of the crystal potential of tungsten between the Fermi level and the vacuum level are d-band states, and are certainly not free-electron-like. In fact, if the surface photoelectric effect is extinguished by
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using normally incident light, structure due to the variation of the density of states of the d-bands in this same energy range is observed in the enhancement factors [31]. This suggests that the final states for surface excitation are the tunnelling states in the vacuum and that the strongest contribution to the surface transition matrix elements comes from the region just outside the metal, where the final density of states is only weakly influenced by the electronic states of the metal. It is not clear from the present experiments whether or not photofield emission can yield information about the final surface density of states of the crystal. It may be that unoccupied surface states and surface resonances which extend into the vacuum can be observed by the present technique (as they can in ultraviolet bremsstrahlung spectroscopy) but that none exist on the particular planes of tungsten and in the energy range studied here. Further experiments with other transition metals and with adsorbed overlayers are required to address this question.
4. Conclusions Photofield emission total energy distributions have been measured from the (100), (110), (111), (211), and (510) crystallographic planes of a tungsten field emitter illuminated by p-polarized light at grazing incidence at seven photon energies between 1.916 and 3.536 eV. Except at the highest photon energy (3.536 eV), the enhancement factors calculated from those distributions have the same form as those previously calculated from field emission distributions, but they are shifted to higher energy by one photon energy. Photofield emission distributions measured at the highest photon energy show departures from the field emission distributions because the photocurrent is emitted above the peak of the surface potential barrier. No evidence has been found for structure due to the surface density of states at the final state energy. These results indicate that the predominant mechanism of excitation by p-polarized light at grazing incidence is surface photoexcitation from occupied crystal states near the surface to free-electron-like tunnelling states in the vacuum. Further experiments are required to determine whether peaks in the final surface density of states due to surface resonances or antibonding states of adsorbed species can be observed through the surface photoelectric effect in photofield emission.
Acknowledgements We are grateful to Paul Donders for many discussions and for his help in solving numerous problems. The excellent technical assistance of K. Weisser of the Scarborough Academic Workshop and J. Shuve of the Electronics
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Workshop are much appreciated. This work was supported in part by operating and equipment grants from the Natural Science and Engineering Research Council (NSERC) of Canada. One of us (DV) wishes to acknowledge financial support from NSERC and from the University of Toronto.
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