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The influence of the electron-phonon scattering on the total energy distribution of field emitted electrons from tungsten
SURFACE
SCIENCE 33 (1972) 589-606 6 North-Holland
THE IN~UEN~ SCATTER~G
Publishing Co.
OF THE E~~RON-P~ONON
ON THE TOTAL
OF FIELD EMITTED
ENERGY
ELECTRONS
JERZY
DISTRIBUTION
FROM TUNGSTEN
J. CZYZEWSKI
institute of Experimentid Physics, University of Wrociaw, Cy~~lskiego 36, Wrochw, Poland
Received 31 March 1972; revised manuscript
received 31 May 1972
The total energy distribution (TED) of field emitted electrons has been measured for [012] direction of tungsten monocrystal in temperature region from 78 K to 9.50 K. To compare measurements with the free electron (FE) theory the parameters of FE equation of TED have been calculated by extrapolation of experimental results to the zero temperature. The explanation of obtained results has been proposed, taking into account the electron scattering phenomena of electrons supplied to the surface during the emission. Characteristic features of s+d and s+s scattering in transition metals described by Mott’s theory have been indicated by TED curves for the [012] direction. The Nottingham effect measured for [012] has been discussed as well.
1. Introduction
Recently, the method of the total energy distribution of field emitted electrons has been used to study the band structure and electron phenomena at the metal surface*~sa-2s). Theoretical investigationss+a2. 2s) show that TED depends considerably on the assumed model of band structure. Experimentally observed anomalies of TED 20*21) of tungsten appear in the [ 1001 crystallographic direction where they are theoretically expected. However, the theorysrss) suggests that the enhancement of emission in the [IOO] direction within some region of energy is caused only by the higher density of energetical states in this region. If one holds the d band responsible for the increase of state density, the consideration of particular transport phenomena in transition metals generates the complication. Namely, the stronger interaction of d electronsa) with ions in a crystal should make the supplying of d electrons to vacanted d states by emission more difficult. In this work we use a new procedure of the comparison of experimental rest&s with the free electron (FE) theory. We assume that the phenomenon of fieId emission from transition metals should not be treated only as the 589
590
J. J. CZYiEWSKI
tunneling of electrons from states of determined energetical structure with the equilibrium distribution of Fermi-Dirac. The existence of two classes of electrons, i.e. free electrons and stronger bound d electrons, seems to give the reasons that one should take into account the temperature dependence of probability of exchange between bands in the transport process to the surface as it was considered in a two-plasma systemss). During the emission, the large d state density and small d electron conductivity should bring about the rise of a large d hole density at the surface for lack of inter-band transitions. But there are conditions that the electron-phonon interaction described by Mott 6 lo) causes s+d transitions and redistribution in the s band throughout s+s transitions. From the angular distribution of d electrons in tungsten crystal*) we can expect that the emission in the [OlZ] direction occurs mainly from the s band. The temperature dependences of the above-mentioned scattering phenomena are in good agreement with the experimental results. At zero temperature the scattering ought to disappear and the Sommerfeld model should be right. At 20 K, Lea and Gomerss) did not observe discrepancies from the FE model below the Fermi level. On this base, we assume that the extrapolation of measurements to zero temperature is a reasonable method of comparison with the FE theory. 2. Theoretical consideration 2.1. THE SOMMERFELD
FREE-ELECTRON MODEL
Most often the experimental TED datars>ssp 12) are compared to Young’s theoryl) based on the free-electron model. The following assumptions are made in this theory: (i) Fermi-Dirac statistics of electrons arriving at the surface of metal. (ii) Classical image model of the surface potential. (iii) Formal expression of TED : E
J(E)dE
=
J(W,E)D(W)dWdE, s -WZ.
(1)
where the supply function J( W, E) gives the number of electrons of energy within the range E to E+dE with z-components of energy W within the range W+ d W incident on the surface z = 0 per unit area per unit time. D(W) is the probability of penetration of the barrier by an electron with z-component of energy W. In this model - it must be emphasized - the free electron gas is understood as if it were at equilibrium in spite of the emission. One obtains the well-
ELECTRON-PHONONSCA’ITERING
ANDTOTALENERGYDISTRIBUTION
591
known expression for theoretical TED :
Jo exp (44 d El + exp (4~41’
J(E) =
(2)
where p = kT/d, E=E-EF, J,, =
(2a) (2b)
e3F2
87r1i(ptZ (v) exp
(24
-
d = AeF/2 (2mq~)%t(v) ,
(W
EF is the Fermi energy, F is the applied field, m is the mass of the electron, 9 is the work function, e is the charge of the electron, and t(r) and v(y) are slowly varying functions 2). Using eq. (2) for a single plane, we consider 40to be constant and then the parameter d is constant too, if the applied field is not changed. In this case the parameter p can be determined by the tip temperature. A position of the maximum of the function (2), E,,,, is determined by both parametersp and d: Emax.= -pdln(l/p-1).
(3)
Also, the expression for the half width of TED at zero temperature is very simple : d(O)=dln+. (4) 2.2
THE
NON-EQUILIBRIUM
SUPPLY
FUNCTION
The formulation of TED introducing the non-equilibrium distribution function has been discussed by Engle and Cutlers). They used Lehovec’sd) theory and Wilson’s5) expression for the relaxation time in the electronlattice interaction. Their non-equilibrium distribution function has the following form : 1 + BJP (W +
K)* g f,,
exP
(‘2 >I.
(5)
B
where B is constant, J is the normal emission current density, and p is the resistivity of the metal. Since J( W,E)=47cm/h3f, according to Engle and Cutlers) one obtains J(E)dE
1 +BJp(W+
=
w,,t$q
-W,
D(W)dWdE.
(6)
592
J. J. CZYiEWSKI
It can be noted in eq. (6) that the nearly free electron model differs from the free electron model by the correction term in brackets. This term is expected to be equal to zero at zero temperature, which means the disappearing of the electron lattice interaction. At constant temperature the difference between J(E) calculated from both models changes monotonically with energy E. In the case of tungsten, the disagreement of the experimental results with the free electron model seems too large and non-monotonic in some region of energy to explain it by means of the non-equilibrium distribution function. Especially, the hump in the total energy distribution of electrons emitted from the [OOl] direction evidences that multi-band effects can not be ignored here. 2.3. THE ELECTRON-PHONONSCATTERINGWITH THE s-+d AND S+S TRANSITIONS Because certain electrical properties of the transition metals were explained by the transitions of the conduction electrons to the unoccupied d states6), they will be used to interpret some total energy distribution measurements. The scattering probabilities depend on the squares of matrix elements of the type :
where AV is the phonon perturbing potential, $k and $k, are the states of the electron before and after scattering. The transition of an electron from the s band can occur to the state in the d band and also to the d band surface state or to the other state in the s band, i.e. there are two processes one when the final ($k,) state lies in the d band and one when it lies in the s band. The matrix element varies with the scattering angle. This problem was discussed by Ziman7) for a monovalent free electron system. In tungsten there is a special situation. It can be distinguished that the [OOl] axis is a direction for scattered s electrons to the d band, because according to Gadzuks) the square of the d electron wave function has a strong maximum in [OOl] of tungsten crystal. The direction perpendicular to the studied single face will be identified at first approximation with the direction of incident electrons. It is motivated by the maximal probability of penetration of the barrier in the direction perpendicular to the surface studied. It seemed reasonable to suppose that the scattering probability is a decreasing function of the scattering angle 0 between [OOI] and incident electrons similarly to the free electron system ‘). The next characteristic virtue of electron scattering is that the transition probabilities for the mentioned two processes are proportional to the densi-
ELECTRON-PHONON
ties of states
N,(E)
SCATTERING
and N,(E)s).
AND TOTAL
ENERGY
It is the problem
DISTRIBUTION
593
which will be mainly
discussed here. In transition metals the density of d states is much larger than the density of the s states. For this reason the high intensity of transitions from s band to d band in the energetical region of high value of N,(E) causes the relaxation time r, to be much less than that calculated from the Wilsons) theory. In the nearly free electron model only s+s transitions are considered actually. Gadzuk’s theorys) of field emission from transition metals considers the tunneling both from d and s band. In some cases the total energy distribution (TED) can be expressed as the sum : J
(El
=
Js
(El
+
Jd
(El
63)
of energy distributions of electrons emitted from s and d bands. The conduction of d electrons is made rather difficult by the strong interaction with ions in the tungsten crystals). The tunneling of electrons from d states localized near the surface can cause the non-equilibrium distribution in the d band which can be decreased by scattering of s electrons to these d states unoccupied after the tunneling. In accordance with the angular distribution of d electrons in tungsten *), the above argumentation explains the enhancement of the emission from the [OOl] direction in some region of energy. On the other hand, it ought to reduce the emission of s electrons in the same region of energy. This fact can be described by the theory of Mott6). The probability of scattering of the s electron by the phonon is given by: P = l/z,, where rs is the time of relaxation ing equation :
for s electrons.
(9 Mott 6310) derived the follow-
where A’ and B depend on the wave function of the incident and scattered electrons and 0 is the Debye temperature. TED of field emitted electrons is studied near the Fermi energy. So, in first aproximation, we can assume that N,(E) = constant compared to N,(E) which varies rapidly with the energy. Near the Fermi energy one obtains: 1 -=G2 [A +BN,(E)]. 7,
(11)
We may note from (11) that the s electron of energy E can be scattered with a probability proportional to the value of the density of d states for the same energy E. This theoretical result of Motts, lo) ought to be reflected in
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J. J. CZYiEWSKI
the TED of s electrons in this energy region where the d band overlaps with the s band. Namely, the density of field emitted s electrons ought to be less than the predictions from the free electron model. This effect should increase with the enhancement of temperature. From the above discussion, one can deduce that in the formal expression of TED (1) the supply function J( W,E) should be multiplied by the probability of non-scattering for an s electron of energy E, which can be expressed by: -W. J,(E)
dE =
J(W,E)[l
- l/r(E)]D(W)dWdE,
(12)
s
W=E
where the sum of both probabilities normalized to the unity :
(of scattering
Pnon-scat.+ l/z(E)
= 1.
The formal equation (12) yields the following emitted s electrons from the transition metal: J, (E) =
(Jo/d) f (E, T) 1 -
&
and non-scattering)
expression
is (13)
for TED of field
(A + BN, (El) 1
eEld.
(14)
We may note from (14) that the discrepancy of the current density from the free electron model increases linearly with temperature and it disappears for T= 0. The next divergence is the reduction of the current density in the region of the existence of the d band and especially of the d state density maximum. In the next section of this paper we shall show that the comparison of the experimental data with the free electron model gives discrepancies that are in qualitative agreement with the proposed model based on the theory of Mott 6, 10). 3. Experimental techniques and procedures 3.1. FIELD EMISSION SPECTROMETER The field emission spectrometer utilized in this study has been described recentlyll) in detail. The geometry of the energy-analyzer electrodes was similar to that used by Van Oostromls), but the hemispherical collector was replaced by a hemispherical gridllpls), which permitted the detection of the electron current by means of an electron multiplier. The multiplier was of the Lallemandia) type. An electrostatic or magnetic method of electron beam deflection causes variation of geometrical conditions for different crystal faces from an electron-optical viewpoint. It is important to keep constant the angle between the probed electron beam and the analyzer axis. For this reason, a mechanical arrangement the emitter was held in a Cardan
ELECTRON-PHONON
SCAlTERING
AND
TOTAL
ENERGY
DISTRIBUTION
595
suspension 15).The deflection angIe could be varied within the range - 45” to -1-45”.Arrangements oftbeemitterand theelectrodes are shown in figs. 1 and2. TED curves were directly measured by the automatic differentiating circuit xi91’). Fig. 3 shows a block diagram of the electrical system. A multiplier output was connected with a dc preamplifier by a 1.5 m doubly shielded cable. Both the internal shield and the preamplifier were at the multiplier anode potential. The carefully shielded multiplier output considerably reduced the noise. The main source of 50 and 100 Hz noise was a high voltage supply of the electron multiplier. Therefore special efforts were made to decrease the ripples and fluctuations of this supply. The batter-supplied
Fig. 1. Schematic diagram of the emitter assembly: (1) the emitter, (2) current potential connections, (3) two-bearing suspension, (4) actuating rod, (5) bellows.
and
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J. J. CZYiEWSKI
Fig. 2, Total energy distribution analyzer: (1) the ceramic rod, (2) anode, (3) lens electrode, (4) shielding electrode, (5) quartz isolator, (6) hemispherical grid, (7) output grid, (8) first dynode of electron multiplier.
preamplifier was assembled in an equipotential box operating at high potential. Circuits behind the preamplifier were isolated from the high voltage
by a 1000 pF capacitor. After ampli~cation by the preampIifier the ac signal passed a selective microvoltmeter (UNIPAN type 227) of 50 dB octave selectivity and then a homodyne rectifier voltmeter (UNIPAN type 202B). The latter was driven by a reference signal obtained from the same oscillator which supplied the modulation signal to the emitter. Finally, the output of the homodyne rectifier voltmeter was coupled down to an xy recorder. The magnitude of the ac signal was plotted on this xy recorder as a function of the retarding voltage. Before sealoff the tube electrodes were outgassed by rf heating and the entire tube was evacuated and baked by the usual procedure. During the operation the entire tube was submerged in liquid nitrogen, which easy enables one to obtain a vacuum in the 10-l’ torr range. Emitter temperatures were controlled by the temperature controller described by Zimmerman and Gomerls). The accuracy oft hese measurements was better than 1”.
ELECTRON-PHONON
SCATTERING
ANDTOTALENERGY DISTRIBUTION
597
Attenuator
Fig. 3. Block diagram of the field emissionspectrometer. 3.2. PROCEDUREAND DATA The tungsten emitter was heated for cleaning graphical plane to be studied was positioned of the Cardan suspension. A [llO]
and then the crystallo-
over the probe hole by means
emitter orientation
was employed.
We
present measurements of TED for the [012] direction at temperatures within the range 78-950 K. The typical set of seventeen TED curves was obtained at different
temperatures
under the constant potentials
of all electrodes
except the retarding one. We assumed that the observed discrepancies were caused by the electronphonon scattering. As it results from Mott’s
theoryS*lO), the free electron
path for this phenomenon increases to infinity when the temperature reaches zero. For this reason - which was discussed in the previous section - the free electron model ought to be the asymptotical one for the emission in the [012]
598
J. J. CZYiEWSKI
direction of tungsten monocrystalss). In consequence of the above assumption, the extrapolation to zero temperature of the following dependences was done: (i) the half width of TED, A versus T; (ii) the peak height of TED, [di,/dV,],,, versus T. The second extrapolation gives the relation: JFE (s,,,)T=o where
E max
= n [di,/dl/,];,=,
is defined by eq. (3), n is a normalization
,
(15)
factor, and
is the TED peak height extrapolated to zero temperature from the experimental data. Generally the TED height at zero temperature normalizes to the unity: JFE(E,,,)==’
=
1
Hence from (16) and (15) 1 It = ~]Z-.
*
(17)
Eq. (17) permits one to calculate the normalization factor for the whole set of TED experimental curves by the evaluation of it at the point where we expect that the discrepancy between FE theory and experimental data is smallest. The variation of the TED height versus T after the normalization is shown in fig. 4.
Fig. 4. Experimentally observed variation of relative peak height J(E& of the total energy distribution curves as a function of p. The dashed line is based on the free electron model, eqs. (2) and (3).
EJ_ECTRON-PHOTON
SCATTERING
AND
TGTAL
ENERGY
DISTRIBIJTXON
599
The calculationof thedparameter for the set of TEDcurves was based on the extrapolation to zero temperature of the temperature variation of the TED half width A. The value of the TED half width d(0) at zero temperature determined by the extrapolation method from the experimental curve A versus T gives the value of the d parameter from eq. (4). Both experimental curves of TED half widths before and after normalization plotted as a function of temperature or p [see eq. (2)] are shown in fig. 5. The value of d(O) determined from fig. 5a is equal 0.086 eV and the value of the parameter d of the presented TED curves resulting from eq. (4) is 0.124 eV. For direction 10121the discrepancy between the theoretical (FE) and experimental curves of TED height versus T (fig. 4) at 78 K is small. Therefore the position of the Fermi energy (&/d=O) for the set of curves was assigned
b
Fig. 5. Variation of half widths A of the experimental energy distribution curves: (a) before normalization as a function of temperature, and (b) after normalization as a function ofp. The dashed line is based on the free electron model, eq. (2).
600
J. J. CZYiEWSKt
at the value of the retarding potential when the ordinate of the normalized TED curve measured at 78 K J(0) is equal to ). Now, we can compare the experimental and theoretical (FE) TED curves at different temperatures (p). In figs. 6-10 there are presented five from fourteen curves of the typical set for the temperatures in the region: 78950 K. The experimental curve for the [OOl] direction measured at 78 K is added in fig. 6 for better illustration of s-td transitions. The investigation of TED curves from the viewpoint of their symmetry yields the value ri = 500 K when both parts of the TED curve on each side
c .a
-a
-a
0
I
E/d
Fig. 6. Experimental total energy distribution plots: (1) along the [0121 direction of the tungsten emitter, where d= 0.124 eV and p = 0.056; (2) along the [OOl] direction of the tungsten emitter, where d= 0.118 eV and p = 0.056. The dashed line is based on the free electron model, eq. (2), where p = 0.056.
ELECTRON-PHONON
/’ __--4
SCATTERING
AND TOTAL
ENERGY
601
DISTRIBUTION
\
/’
\ \ \
_** -3
-2
0
1
E/d
Fig. 7. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d =0.124eV and p =0.21. The dashed line is based on the free electron model, where p = 0.21.
of its maximum have been symmetrical. However, the axis of symmetry does not cover the Fermi energy axis (fig. 8). The displacement amounts to 0.012eVwhichismuch more than the experimental error. Also, the comparison of the obtained experimental value Ti = 500 K with the theoretical temperature of inversion in the Nottingham effect Tr = 720 K for d= 0.124 eV gives a large disagreement. Swanson, Crouser and Charbonniera’) observed a discrepancy in the inversion temperature of the same nature. The effect of the quantum-modified potential barrier proposed by Engle and Cutler 24) yields a correction to the amount of 8” in relation to the classical image potential and the free electron model. The typical variation of the monoenergetic electron current with temperature (p) is illustrated in fig. 11. According to the free electron model, the emission J(0) from the Fermi level should be constant (the dashed line in fig. 11) with the variation of temperature(p). The experimental data (the solid
602
J. J.
CZYiEWSKI
J
w .6
,,---. I
.s’,
‘\
I
1)
/I I
\
\\
A
\
/I’ / I/
\
\ \ \
\
.3
I /
/’
I/
/’
.e
J
I’ I’
/ /’
0.1
I
__e-
*-
--
c’
NH’ ‘.
-’ -3
-.
4 -1
-8
0
I
f/d
Fig. 8. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d = 0.124 eV and p = 0.35. The dashed line is based on the free electron model, where p = 0.35.
line
in fig. 11) show a near
linear
decrease
of J(0) at high temperatures.
4. Discussion The comparison of experimental results with the FE model - which was done in the previous section - shows that there are serious discrepancies also for other directions than [OOl]. Briefly we can emphasize the following observations : (i) At constant temperature - it can be noted especially at 78 K (fig. 6) - the shape of the TED curve for [012] does not fit the FE theory, particularly in the energetical region where the enhancement of the emission current density for [OOl] exists. But in the case of [012], the discrepancy is in the direction of the lower current density as related to FE theory. In other words, there is the strongest reduction of emission in [012] within the same range of energy where we observe the hump in the [OOl] direction20*21). (ii) If one investigates the family of TED curves (figs. 6-lo), it can be noted that the maximum reaches the position s/d=0 for a value of p lower than 3
ELECTRON-PHONON
-3
-2
SCATTERING
-1
AND TOTAL
ENERGY
0
603
DISTRIBUTION
I
w
Fig. 9. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d= 0.12 eV and p = 0.50. The dashed line is based on the free electron model, where p = 0.50.
and than predicted from FE theory. Fig. 8 shows that the maximum of TED is already placed on the side of positive energies for p = +. (iii) The experimental variation of the relative TED height with the temperature has a different character than expected from the FE model (fig. 5). Neither of the observations mentioned above can be explained by means of the effect of band structure only *~aa~as).In principle, on the basis of the proposed theory** 221ss), the enhancement of emission in the energetical region of the hump should be observed up to 90”. It can rapidly decrease to the FE level indeed but not below it (i). The direction [012] is only turned through the angle 26” 34’ with respect to [OOl]. However, we observed (i) the significant reduction of emission instead of even considerable enhancement. It can be noted in fig. 6 that the [OOl] hump and the [012] reduction of emission are of the same order of magnitude. The explanation seems reasonable on the basis of our consideration in section 2. At first, we assume for [012] that the contribution of J,(E) in eq. (8) is not significant and the emission occurs from the s band in the first place. Eq. (14) gives the dependence of emission on the energy and the temperature, taking into account
604
J. J. CZYiEWSKI
the s+d transitions. The expression in brackets can be much less than the unity for the energetical region of the high density of d states. At 78 K the shape of the TED curve (fig. 6) qualitatively agrees with the above condition. The region of the [OOl] hump corresponds with the high density of
-1
0
1
2
Efd
Fig. 10. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d = 0.124 eV and p = 0.66. The dashed line is based on the free electron model, where p = 0.66.
u! 0 Fig. 11.
0.2
0,4
0.6
P
Experimentally observed variation of the monoenergetic electron density J(0) along the [012] direction of the tungsten emitter from the Fermi level as a function of p.
current
ELECTRON-PHONON
the d states. The emission
SCATTERING
AND TOTAL
of s electrons
ENERGY
throughout
DISTRIBUTION
605
the d states in the pro-
cess of s-d scattering empties the s states in some limited region of energy. In all regions of energy the redistribution of the s electrons is caused by the process of s+s scattering. In (14) this fact is described by the term:
proportional to the temperature. Experimental data presented in fig. 11 confirm this dependence of emission current density on the temperature in the region above of the Debye temperature. The deviation from the FE model is a linear function of the parameter p above the value 0.2. The maximum of the curve in fig. 11 is interpreted by some contribution of the term J,(E) in eq. (8) which can still not be neglected at low temperatures even for [012]. This problem will be separatly discussed in the next part of this paper more exactly. The measurements of the author yield the value EJd= - 3.05 on the abscissa, where the TED curve in [OOl] has the maximal deviation from the TED curve calculated from the FE model, which is identified with the position of the maximum of the d state densityss20r2s). When we consider the TED curve from the viewpoint of the position of its maximum (ii), including the scattering processes, the displacement of the experimental [012] TED maximum to the right in relation to the FE maximum of TED can be easy explained. Nearer by Qd= -3.05 the s+d scattering is stronger and the reduction of emission of s electrons is greater. The more distinct lowering of the left side of the TED curves in figs. 6-10 causes the displacement of the position of the TED maximum to the right. The obtained symmetrical curve of [012] TED for p=O.35 and the displacement of the axis of symmetry in relation to the Fermi energy is due to the same mechanism as described above. The character of the experimental TED peak height with the temperature in relation to that variation calculated from the FE model (fig. 5) is similar to the dependence of J(0) on p. However, the position of TED changes at different temperatures going to the energetical region of larger d state density at first and then of smaller d state density at the highest temperatures. In the FE model, this fact is described by eq. (3). It is the reason that the interpretation is more complicated. Both the variation JJe,,J versus p and the contribution of J.(E,) versus p are not linear. The qualitative calculation of the supply function in the total energy distribution of electrons emitted from tungsten requires the taking into account of both collectives of electrons: d and s as well as the transitions
606
J. J.
CZYiEWSKI
s+d and s-s. It seems that the fuller theoretical description should be based on the theory including the particular angular distribution of d electrons in tungsten single crystal and phenomena of scattering as well.
Acknowledgments The author acknowledgements the critical reading of the manuscript Dot. Z. Sidorski and Dr. J. W. Gadzuk from the National Bureau Standards.
by of
References 1) R. D. Young, Phys. Rev. 113 (1959) 110. 2) R. H. Good and E. W. Mtiller, in: Hundbuch der Physik, Vol. 21 Ed. S. Fliigge, (Springer, Berlin, 1956). 3) Irene M. Engle and P. H. Cutler, Surface Sci. 12 (1968) 208. 4) K. Lehovec, Phys. Rev. 96 (1954) 921. 5) A. H. Wilson, The Theory of Metals (Cambridge Univ. Press, 1965) sections 93, 95. 6) N. F. Mott Proc. Roy. Sot. (London) A 153 (1936) 699. 7) J. M. Ziman, Electrons andf’honons (Oxford, 1960)~. 187 and fig. 66. 8) J. W. Gadzuk, Phys. Rev. 182 (1969) 416. 9) B. Rozenfeld, Acta Phys. Polonica 31(1967) 197. 10) N. F. Mott, Proc. Roy. Sot. (London) A 156 (1936) 368. 11) C. Workowski and J. J. Czyiewski, Acta Phys. Polonica A 39 (1971) 523. 12) A. G. J. van Oostrom, Philips Res. Rept. Suppl. No. 1 (1966). 13) A. R. L. Moss and I. B. H. Blott, Surface Sci. 17 (1969) 240. 14) A. Lallemand, Vide 21(1949) 618. 15) L. D. Schmidt and R. Gomer J. Chem. Phys. 45 (1966) 1605. 16) A. M. Russell, Rev. Sci. Instr. 33 (1962) 1324. 17) L. B. Leder and J. A. Simpson, Rev. Sci. Tnstr. 29 (1958) 571. 18) D. Zimmerman and R. Gomer, Rev. Sci Instr. 36 (1965) 1046. 19) R. D. Young and E. W. Mtiller, Phys. Rev. 113 (1959) 115. 20) L. W. Swanson and L. C. Crouser, Phys. Rev. 163 (1967) 622. 21) E. W. Plummer and R. D. Young, Phys. Rev. B l(l970) 2088. 22) R. Stratton, Phys. Rev. 135 (1964) A794. 23) D. Nagy and P. H. Cutler, Phys. Rev. 186 (1969) 651. 24) Irene M. Engle and P. H. Cutler, Surface Sci. 8 (1967) 288. 25) D. Pines, Can. J. Phys. 34 (1956) 1379. 26) C. Lea and R. Gomer, Phys. Rev. Letters 25 (1970) 804. 27) L. W. Swanson, L. C. Crouser and F. M. Charbonnier, Phys. Rev. 151 (1966) 327.
SCIENCE 33 (1972) 589-606 6 North-Holland
THE IN~UEN~ SCATTER~G
Publishing Co.
OF THE E~~RON-P~ONON
ON THE TOTAL
OF FIELD EMITTED
ENERGY
ELECTRONS
JERZY
DISTRIBUTION
FROM TUNGSTEN
J. CZYZEWSKI
institute of Experimentid Physics, University of Wrociaw, Cy~~lskiego 36, Wrochw, Poland
Received 31 March 1972; revised manuscript
received 31 May 1972
The total energy distribution (TED) of field emitted electrons has been measured for [012] direction of tungsten monocrystal in temperature region from 78 K to 9.50 K. To compare measurements with the free electron (FE) theory the parameters of FE equation of TED have been calculated by extrapolation of experimental results to the zero temperature. The explanation of obtained results has been proposed, taking into account the electron scattering phenomena of electrons supplied to the surface during the emission. Characteristic features of s+d and s+s scattering in transition metals described by Mott’s theory have been indicated by TED curves for the [012] direction. The Nottingham effect measured for [012] has been discussed as well.
1. Introduction
Recently, the method of the total energy distribution of field emitted electrons has been used to study the band structure and electron phenomena at the metal surface*~sa-2s). Theoretical investigationss+a2. 2s) show that TED depends considerably on the assumed model of band structure. Experimentally observed anomalies of TED 20*21) of tungsten appear in the [ 1001 crystallographic direction where they are theoretically expected. However, the theorysrss) suggests that the enhancement of emission in the [IOO] direction within some region of energy is caused only by the higher density of energetical states in this region. If one holds the d band responsible for the increase of state density, the consideration of particular transport phenomena in transition metals generates the complication. Namely, the stronger interaction of d electronsa) with ions in a crystal should make the supplying of d electrons to vacanted d states by emission more difficult. In this work we use a new procedure of the comparison of experimental rest&s with the free electron (FE) theory. We assume that the phenomenon of fieId emission from transition metals should not be treated only as the 589
590
J. J. CZYiEWSKI
tunneling of electrons from states of determined energetical structure with the equilibrium distribution of Fermi-Dirac. The existence of two classes of electrons, i.e. free electrons and stronger bound d electrons, seems to give the reasons that one should take into account the temperature dependence of probability of exchange between bands in the transport process to the surface as it was considered in a two-plasma systemss). During the emission, the large d state density and small d electron conductivity should bring about the rise of a large d hole density at the surface for lack of inter-band transitions. But there are conditions that the electron-phonon interaction described by Mott 6 lo) causes s+d transitions and redistribution in the s band throughout s+s transitions. From the angular distribution of d electrons in tungsten crystal*) we can expect that the emission in the [OlZ] direction occurs mainly from the s band. The temperature dependences of the above-mentioned scattering phenomena are in good agreement with the experimental results. At zero temperature the scattering ought to disappear and the Sommerfeld model should be right. At 20 K, Lea and Gomerss) did not observe discrepancies from the FE model below the Fermi level. On this base, we assume that the extrapolation of measurements to zero temperature is a reasonable method of comparison with the FE theory. 2. Theoretical consideration 2.1. THE SOMMERFELD
FREE-ELECTRON MODEL
Most often the experimental TED datars>ssp 12) are compared to Young’s theoryl) based on the free-electron model. The following assumptions are made in this theory: (i) Fermi-Dirac statistics of electrons arriving at the surface of metal. (ii) Classical image model of the surface potential. (iii) Formal expression of TED : E
J(E)dE
=
J(W,E)D(W)dWdE, s -WZ.
(1)
where the supply function J( W, E) gives the number of electrons of energy within the range E to E+dE with z-components of energy W within the range W+ d W incident on the surface z = 0 per unit area per unit time. D(W) is the probability of penetration of the barrier by an electron with z-component of energy W. In this model - it must be emphasized - the free electron gas is understood as if it were at equilibrium in spite of the emission. One obtains the well-
ELECTRON-PHONONSCA’ITERING
ANDTOTALENERGYDISTRIBUTION
591
known expression for theoretical TED :
Jo exp (44 d El + exp (4~41’
J(E) =
(2)
where p = kT/d, E=E-EF, J,, =
(2a) (2b)
e3F2
87r1i(ptZ (v) exp
(24
-
d = AeF/2 (2mq~)%t(v) ,
(W
EF is the Fermi energy, F is the applied field, m is the mass of the electron, 9 is the work function, e is the charge of the electron, and t(r) and v(y) are slowly varying functions 2). Using eq. (2) for a single plane, we consider 40to be constant and then the parameter d is constant too, if the applied field is not changed. In this case the parameter p can be determined by the tip temperature. A position of the maximum of the function (2), E,,,, is determined by both parametersp and d: Emax.= -pdln(l/p-1).
(3)
Also, the expression for the half width of TED at zero temperature is very simple : d(O)=dln+. (4) 2.2
THE
NON-EQUILIBRIUM
SUPPLY
FUNCTION
The formulation of TED introducing the non-equilibrium distribution function has been discussed by Engle and Cutlers). They used Lehovec’sd) theory and Wilson’s5) expression for the relaxation time in the electronlattice interaction. Their non-equilibrium distribution function has the following form : 1 + BJP (W +
K)* g f,,
exP
(‘2 >I.
(5)
B
where B is constant, J is the normal emission current density, and p is the resistivity of the metal. Since J( W,E)=47cm/h3f, according to Engle and Cutlers) one obtains J(E)dE
1 +BJp(W+
=
w,,t$q
-W,
D(W)dWdE.
(6)
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J. J. CZYiEWSKI
It can be noted in eq. (6) that the nearly free electron model differs from the free electron model by the correction term in brackets. This term is expected to be equal to zero at zero temperature, which means the disappearing of the electron lattice interaction. At constant temperature the difference between J(E) calculated from both models changes monotonically with energy E. In the case of tungsten, the disagreement of the experimental results with the free electron model seems too large and non-monotonic in some region of energy to explain it by means of the non-equilibrium distribution function. Especially, the hump in the total energy distribution of electrons emitted from the [OOl] direction evidences that multi-band effects can not be ignored here. 2.3. THE ELECTRON-PHONONSCATTERINGWITH THE s-+d AND S+S TRANSITIONS Because certain electrical properties of the transition metals were explained by the transitions of the conduction electrons to the unoccupied d states6), they will be used to interpret some total energy distribution measurements. The scattering probabilities depend on the squares of matrix elements of the type :
where AV is the phonon perturbing potential, $k and $k, are the states of the electron before and after scattering. The transition of an electron from the s band can occur to the state in the d band and also to the d band surface state or to the other state in the s band, i.e. there are two processes one when the final ($k,) state lies in the d band and one when it lies in the s band. The matrix element varies with the scattering angle. This problem was discussed by Ziman7) for a monovalent free electron system. In tungsten there is a special situation. It can be distinguished that the [OOl] axis is a direction for scattered s electrons to the d band, because according to Gadzuks) the square of the d electron wave function has a strong maximum in [OOl] of tungsten crystal. The direction perpendicular to the studied single face will be identified at first approximation with the direction of incident electrons. It is motivated by the maximal probability of penetration of the barrier in the direction perpendicular to the surface studied. It seemed reasonable to suppose that the scattering probability is a decreasing function of the scattering angle 0 between [OOI] and incident electrons similarly to the free electron system ‘). The next characteristic virtue of electron scattering is that the transition probabilities for the mentioned two processes are proportional to the densi-
ELECTRON-PHONON
ties of states
N,(E)
SCATTERING
and N,(E)s).
AND TOTAL
ENERGY
It is the problem
DISTRIBUTION
593
which will be mainly
discussed here. In transition metals the density of d states is much larger than the density of the s states. For this reason the high intensity of transitions from s band to d band in the energetical region of high value of N,(E) causes the relaxation time r, to be much less than that calculated from the Wilsons) theory. In the nearly free electron model only s+s transitions are considered actually. Gadzuk’s theorys) of field emission from transition metals considers the tunneling both from d and s band. In some cases the total energy distribution (TED) can be expressed as the sum : J
(El
=
Js
(El
+
Jd
(El
63)
of energy distributions of electrons emitted from s and d bands. The conduction of d electrons is made rather difficult by the strong interaction with ions in the tungsten crystals). The tunneling of electrons from d states localized near the surface can cause the non-equilibrium distribution in the d band which can be decreased by scattering of s electrons to these d states unoccupied after the tunneling. In accordance with the angular distribution of d electrons in tungsten *), the above argumentation explains the enhancement of the emission from the [OOl] direction in some region of energy. On the other hand, it ought to reduce the emission of s electrons in the same region of energy. This fact can be described by the theory of Mott6). The probability of scattering of the s electron by the phonon is given by: P = l/z,, where rs is the time of relaxation ing equation :
for s electrons.
(9 Mott 6310) derived the follow-
where A’ and B depend on the wave function of the incident and scattered electrons and 0 is the Debye temperature. TED of field emitted electrons is studied near the Fermi energy. So, in first aproximation, we can assume that N,(E) = constant compared to N,(E) which varies rapidly with the energy. Near the Fermi energy one obtains: 1 -=G2 [A +BN,(E)]. 7,
(11)
We may note from (11) that the s electron of energy E can be scattered with a probability proportional to the value of the density of d states for the same energy E. This theoretical result of Motts, lo) ought to be reflected in
594
J. J. CZYiEWSKI
the TED of s electrons in this energy region where the d band overlaps with the s band. Namely, the density of field emitted s electrons ought to be less than the predictions from the free electron model. This effect should increase with the enhancement of temperature. From the above discussion, one can deduce that in the formal expression of TED (1) the supply function J( W,E) should be multiplied by the probability of non-scattering for an s electron of energy E, which can be expressed by: -W. J,(E)
dE =
J(W,E)[l
- l/r(E)]D(W)dWdE,
(12)
s
W=E
where the sum of both probabilities normalized to the unity :
(of scattering
Pnon-scat.+ l/z(E)
= 1.
The formal equation (12) yields the following emitted s electrons from the transition metal: J, (E) =
(Jo/d) f (E, T) 1 -
&
and non-scattering)
expression
is (13)
for TED of field
(A + BN, (El) 1
eEld.
(14)
We may note from (14) that the discrepancy of the current density from the free electron model increases linearly with temperature and it disappears for T= 0. The next divergence is the reduction of the current density in the region of the existence of the d band and especially of the d state density maximum. In the next section of this paper we shall show that the comparison of the experimental data with the free electron model gives discrepancies that are in qualitative agreement with the proposed model based on the theory of Mott 6, 10). 3. Experimental techniques and procedures 3.1. FIELD EMISSION SPECTROMETER The field emission spectrometer utilized in this study has been described recentlyll) in detail. The geometry of the energy-analyzer electrodes was similar to that used by Van Oostromls), but the hemispherical collector was replaced by a hemispherical gridllpls), which permitted the detection of the electron current by means of an electron multiplier. The multiplier was of the Lallemandia) type. An electrostatic or magnetic method of electron beam deflection causes variation of geometrical conditions for different crystal faces from an electron-optical viewpoint. It is important to keep constant the angle between the probed electron beam and the analyzer axis. For this reason, a mechanical arrangement the emitter was held in a Cardan
ELECTRON-PHONON
SCAlTERING
AND
TOTAL
ENERGY
DISTRIBUTION
595
suspension 15).The deflection angIe could be varied within the range - 45” to -1-45”.Arrangements oftbeemitterand theelectrodes are shown in figs. 1 and2. TED curves were directly measured by the automatic differentiating circuit xi91’). Fig. 3 shows a block diagram of the electrical system. A multiplier output was connected with a dc preamplifier by a 1.5 m doubly shielded cable. Both the internal shield and the preamplifier were at the multiplier anode potential. The carefully shielded multiplier output considerably reduced the noise. The main source of 50 and 100 Hz noise was a high voltage supply of the electron multiplier. Therefore special efforts were made to decrease the ripples and fluctuations of this supply. The batter-supplied
Fig. 1. Schematic diagram of the emitter assembly: (1) the emitter, (2) current potential connections, (3) two-bearing suspension, (4) actuating rod, (5) bellows.
and
596
J. J. CZYiEWSKI
Fig. 2, Total energy distribution analyzer: (1) the ceramic rod, (2) anode, (3) lens electrode, (4) shielding electrode, (5) quartz isolator, (6) hemispherical grid, (7) output grid, (8) first dynode of electron multiplier.
preamplifier was assembled in an equipotential box operating at high potential. Circuits behind the preamplifier were isolated from the high voltage
by a 1000 pF capacitor. After ampli~cation by the preampIifier the ac signal passed a selective microvoltmeter (UNIPAN type 227) of 50 dB octave selectivity and then a homodyne rectifier voltmeter (UNIPAN type 202B). The latter was driven by a reference signal obtained from the same oscillator which supplied the modulation signal to the emitter. Finally, the output of the homodyne rectifier voltmeter was coupled down to an xy recorder. The magnitude of the ac signal was plotted on this xy recorder as a function of the retarding voltage. Before sealoff the tube electrodes were outgassed by rf heating and the entire tube was evacuated and baked by the usual procedure. During the operation the entire tube was submerged in liquid nitrogen, which easy enables one to obtain a vacuum in the 10-l’ torr range. Emitter temperatures were controlled by the temperature controller described by Zimmerman and Gomerls). The accuracy oft hese measurements was better than 1”.
ELECTRON-PHONON
SCATTERING
ANDTOTALENERGY DISTRIBUTION
597
Attenuator
Fig. 3. Block diagram of the field emissionspectrometer. 3.2. PROCEDUREAND DATA The tungsten emitter was heated for cleaning graphical plane to be studied was positioned of the Cardan suspension. A [llO]
and then the crystallo-
over the probe hole by means
emitter orientation
was employed.
We
present measurements of TED for the [012] direction at temperatures within the range 78-950 K. The typical set of seventeen TED curves was obtained at different
temperatures
under the constant potentials
of all electrodes
except the retarding one. We assumed that the observed discrepancies were caused by the electronphonon scattering. As it results from Mott’s
theoryS*lO), the free electron
path for this phenomenon increases to infinity when the temperature reaches zero. For this reason - which was discussed in the previous section - the free electron model ought to be the asymptotical one for the emission in the [012]
598
J. J. CZYiEWSKI
direction of tungsten monocrystalss). In consequence of the above assumption, the extrapolation to zero temperature of the following dependences was done: (i) the half width of TED, A versus T; (ii) the peak height of TED, [di,/dV,],,, versus T. The second extrapolation gives the relation: JFE (s,,,)T=o where
E max
= n [di,/dl/,];,=,
is defined by eq. (3), n is a normalization
,
(15)
factor, and
is the TED peak height extrapolated to zero temperature from the experimental data. Generally the TED height at zero temperature normalizes to the unity: JFE(E,,,)==’
=
1
Hence from (16) and (15) 1 It = ~]Z-.
*
(17)
Eq. (17) permits one to calculate the normalization factor for the whole set of TED experimental curves by the evaluation of it at the point where we expect that the discrepancy between FE theory and experimental data is smallest. The variation of the TED height versus T after the normalization is shown in fig. 4.
Fig. 4. Experimentally observed variation of relative peak height J(E& of the total energy distribution curves as a function of p. The dashed line is based on the free electron model, eqs. (2) and (3).
EJ_ECTRON-PHOTON
SCATTERING
AND
TGTAL
ENERGY
DISTRIBIJTXON
599
The calculationof thedparameter for the set of TEDcurves was based on the extrapolation to zero temperature of the temperature variation of the TED half width A. The value of the TED half width d(0) at zero temperature determined by the extrapolation method from the experimental curve A versus T gives the value of the d parameter from eq. (4). Both experimental curves of TED half widths before and after normalization plotted as a function of temperature or p [see eq. (2)] are shown in fig. 5. The value of d(O) determined from fig. 5a is equal 0.086 eV and the value of the parameter d of the presented TED curves resulting from eq. (4) is 0.124 eV. For direction 10121the discrepancy between the theoretical (FE) and experimental curves of TED height versus T (fig. 4) at 78 K is small. Therefore the position of the Fermi energy (&/d=O) for the set of curves was assigned
b
Fig. 5. Variation of half widths A of the experimental energy distribution curves: (a) before normalization as a function of temperature, and (b) after normalization as a function ofp. The dashed line is based on the free electron model, eq. (2).
600
J. J. CZYiEWSKt
at the value of the retarding potential when the ordinate of the normalized TED curve measured at 78 K J(0) is equal to ). Now, we can compare the experimental and theoretical (FE) TED curves at different temperatures (p). In figs. 6-10 there are presented five from fourteen curves of the typical set for the temperatures in the region: 78950 K. The experimental curve for the [OOl] direction measured at 78 K is added in fig. 6 for better illustration of s-td transitions. The investigation of TED curves from the viewpoint of their symmetry yields the value ri = 500 K when both parts of the TED curve on each side
c .a
-a
-a
0
I
E/d
Fig. 6. Experimental total energy distribution plots: (1) along the [0121 direction of the tungsten emitter, where d= 0.124 eV and p = 0.056; (2) along the [OOl] direction of the tungsten emitter, where d= 0.118 eV and p = 0.056. The dashed line is based on the free electron model, eq. (2), where p = 0.056.
ELECTRON-PHONON
/’ __--4
SCATTERING
AND TOTAL
ENERGY
601
DISTRIBUTION
\
/’
\ \ \
_** -3
-2
0
1
E/d
Fig. 7. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d =0.124eV and p =0.21. The dashed line is based on the free electron model, where p = 0.21.
of its maximum have been symmetrical. However, the axis of symmetry does not cover the Fermi energy axis (fig. 8). The displacement amounts to 0.012eVwhichismuch more than the experimental error. Also, the comparison of the obtained experimental value Ti = 500 K with the theoretical temperature of inversion in the Nottingham effect Tr = 720 K for d= 0.124 eV gives a large disagreement. Swanson, Crouser and Charbonniera’) observed a discrepancy in the inversion temperature of the same nature. The effect of the quantum-modified potential barrier proposed by Engle and Cutler 24) yields a correction to the amount of 8” in relation to the classical image potential and the free electron model. The typical variation of the monoenergetic electron current with temperature (p) is illustrated in fig. 11. According to the free electron model, the emission J(0) from the Fermi level should be constant (the dashed line in fig. 11) with the variation of temperature(p). The experimental data (the solid
602
J. J.
CZYiEWSKI
J
w .6
,,---. I
.s’,
‘\
I
1)
/I I
\
\\
A
\
/I’ / I/
\
\ \ \
\
.3
I /
/’
I/
/’
.e
J
I’ I’
/ /’
0.1
I
__e-
*-
--
c’
NH’ ‘.
-’ -3
-.
4 -1
-8
0
I
f/d
Fig. 8. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d = 0.124 eV and p = 0.35. The dashed line is based on the free electron model, where p = 0.35.
line
in fig. 11) show a near
linear
decrease
of J(0) at high temperatures.
4. Discussion The comparison of experimental results with the FE model - which was done in the previous section - shows that there are serious discrepancies also for other directions than [OOl]. Briefly we can emphasize the following observations : (i) At constant temperature - it can be noted especially at 78 K (fig. 6) - the shape of the TED curve for [012] does not fit the FE theory, particularly in the energetical region where the enhancement of the emission current density for [OOl] exists. But in the case of [012], the discrepancy is in the direction of the lower current density as related to FE theory. In other words, there is the strongest reduction of emission in [012] within the same range of energy where we observe the hump in the [OOl] direction20*21). (ii) If one investigates the family of TED curves (figs. 6-lo), it can be noted that the maximum reaches the position s/d=0 for a value of p lower than 3
ELECTRON-PHONON
-3
-2
SCATTERING
-1
AND TOTAL
ENERGY
0
603
DISTRIBUTION
I
w
Fig. 9. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d= 0.12 eV and p = 0.50. The dashed line is based on the free electron model, where p = 0.50.
and than predicted from FE theory. Fig. 8 shows that the maximum of TED is already placed on the side of positive energies for p = +. (iii) The experimental variation of the relative TED height with the temperature has a different character than expected from the FE model (fig. 5). Neither of the observations mentioned above can be explained by means of the effect of band structure only *~aa~as).In principle, on the basis of the proposed theory** 221ss), the enhancement of emission in the energetical region of the hump should be observed up to 90”. It can rapidly decrease to the FE level indeed but not below it (i). The direction [012] is only turned through the angle 26” 34’ with respect to [OOl]. However, we observed (i) the significant reduction of emission instead of even considerable enhancement. It can be noted in fig. 6 that the [OOl] hump and the [012] reduction of emission are of the same order of magnitude. The explanation seems reasonable on the basis of our consideration in section 2. At first, we assume for [012] that the contribution of J,(E) in eq. (8) is not significant and the emission occurs from the s band in the first place. Eq. (14) gives the dependence of emission on the energy and the temperature, taking into account
604
J. J. CZYiEWSKI
the s+d transitions. The expression in brackets can be much less than the unity for the energetical region of the high density of d states. At 78 K the shape of the TED curve (fig. 6) qualitatively agrees with the above condition. The region of the [OOl] hump corresponds with the high density of
-1
0
1
2
Efd
Fig. 10. Experimental total energy distribution plot along the [012] direction of the tungsten emitter, where d = 0.124 eV and p = 0.66. The dashed line is based on the free electron model, where p = 0.66.
u! 0 Fig. 11.
0.2
0,4
0.6
P
Experimentally observed variation of the monoenergetic electron density J(0) along the [012] direction of the tungsten emitter from the Fermi level as a function of p.
current
ELECTRON-PHONON
the d states. The emission
SCATTERING
AND TOTAL
of s electrons
ENERGY
throughout
DISTRIBUTION
605
the d states in the pro-
cess of s-d scattering empties the s states in some limited region of energy. In all regions of energy the redistribution of the s electrons is caused by the process of s+s scattering. In (14) this fact is described by the term:
proportional to the temperature. Experimental data presented in fig. 11 confirm this dependence of emission current density on the temperature in the region above of the Debye temperature. The deviation from the FE model is a linear function of the parameter p above the value 0.2. The maximum of the curve in fig. 11 is interpreted by some contribution of the term J,(E) in eq. (8) which can still not be neglected at low temperatures even for [012]. This problem will be separatly discussed in the next part of this paper more exactly. The measurements of the author yield the value EJd= - 3.05 on the abscissa, where the TED curve in [OOl] has the maximal deviation from the TED curve calculated from the FE model, which is identified with the position of the maximum of the d state densityss20r2s). When we consider the TED curve from the viewpoint of the position of its maximum (ii), including the scattering processes, the displacement of the experimental [012] TED maximum to the right in relation to the FE maximum of TED can be easy explained. Nearer by Qd= -3.05 the s+d scattering is stronger and the reduction of emission of s electrons is greater. The more distinct lowering of the left side of the TED curves in figs. 6-10 causes the displacement of the position of the TED maximum to the right. The obtained symmetrical curve of [012] TED for p=O.35 and the displacement of the axis of symmetry in relation to the Fermi energy is due to the same mechanism as described above. The character of the experimental TED peak height with the temperature in relation to that variation calculated from the FE model (fig. 5) is similar to the dependence of J(0) on p. However, the position of TED changes at different temperatures going to the energetical region of larger d state density at first and then of smaller d state density at the highest temperatures. In the FE model, this fact is described by eq. (3). It is the reason that the interpretation is more complicated. Both the variation JJe,,J versus p and the contribution of J.(E,) versus p are not linear. The qualitative calculation of the supply function in the total energy distribution of electrons emitted from tungsten requires the taking into account of both collectives of electrons: d and s as well as the transitions
606
J. J.
CZYiEWSKI
s+d and s-s. It seems that the fuller theoretical description should be based on the theory including the particular angular distribution of d electrons in tungsten single crystal and phenomena of scattering as well.
Acknowledgments The author acknowledgements the critical reading of the manuscript Dot. Z. Sidorski and Dr. J. W. Gadzuk from the National Bureau Standards.
by of
References 1) R. D. Young, Phys. Rev. 113 (1959) 110. 2) R. H. Good and E. W. Mtiller, in: Hundbuch der Physik, Vol. 21 Ed. S. Fliigge, (Springer, Berlin, 1956). 3) Irene M. Engle and P. H. Cutler, Surface Sci. 12 (1968) 208. 4) K. Lehovec, Phys. Rev. 96 (1954) 921. 5) A. H. Wilson, The Theory of Metals (Cambridge Univ. Press, 1965) sections 93, 95. 6) N. F. Mott Proc. Roy. Sot. (London) A 153 (1936) 699. 7) J. M. Ziman, Electrons andf’honons (Oxford, 1960)~. 187 and fig. 66. 8) J. W. Gadzuk, Phys. Rev. 182 (1969) 416. 9) B. Rozenfeld, Acta Phys. Polonica 31(1967) 197. 10) N. F. Mott, Proc. Roy. Sot. (London) A 156 (1936) 368. 11) C. Workowski and J. J. Czyiewski, Acta Phys. Polonica A 39 (1971) 523. 12) A. G. J. van Oostrom, Philips Res. Rept. Suppl. No. 1 (1966). 13) A. R. L. Moss and I. B. H. Blott, Surface Sci. 17 (1969) 240. 14) A. Lallemand, Vide 21(1949) 618. 15) L. D. Schmidt and R. Gomer J. Chem. Phys. 45 (1966) 1605. 16) A. M. Russell, Rev. Sci. Instr. 33 (1962) 1324. 17) L. B. Leder and J. A. Simpson, Rev. Sci. Tnstr. 29 (1958) 571. 18) D. Zimmerman and R. Gomer, Rev. Sci Instr. 36 (1965) 1046. 19) R. D. Young and E. W. Mtiller, Phys. Rev. 113 (1959) 115. 20) L. W. Swanson and L. C. Crouser, Phys. Rev. 163 (1967) 622. 21) E. W. Plummer and R. D. Young, Phys. Rev. B l(l970) 2088. 22) R. Stratton, Phys. Rev. 135 (1964) A794. 23) D. Nagy and P. H. Cutler, Phys. Rev. 186 (1969) 651. 24) Irene M. Engle and P. H. Cutler, Surface Sci. 8 (1967) 288. 25) D. Pines, Can. J. Phys. 34 (1956) 1379. 26) C. Lea and R. Gomer, Phys. Rev. Letters 25 (1970) 804. 27) L. W. Swanson, L. C. Crouser and F. M. Charbonnier, Phys. Rev. 151 (1966) 327.