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Field emission work functions
Surface Science 48 (1975) 417-431 © North-Holland Publishing Company
FIELD EMISSION WORK FUNCTIONS T.V. V O R B U R G E R a n d D. P E N N
National Bureau of Standards, Washington, D.C. 20234, U.S.A. and E.W. P L U M M E R
Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A. Received 16 July 1974; manuscript received in final form 25 October 1974
Field emission has proven to be a very useful technique for obtaining work function changes from single crystal planes or from the whole emitter. The inability to independently measure the electric field has limited the accuracy of field emission total current measurements for determining absolute work functions. Young and Clark's method of combining field emission energy distribution measurements and total current versus voltage measurements to eliminate the electric field as an unknown is not adequate because it does not take into account the effects of the surface density of states present in the energy distribution. In this paper we discuss a technique to overcome this problem, which involves a series of measurements taken as a function of electric field and an extrapolation to zero field. The method yields reliable values of the work function for the low index (fiat) (100) and (112) planes of tungsten but not for the high index (rough) (013) and (111) faces.
I. i n t r o d u c t i o n T h e c u r r e n t - v o l t a g e characteristics o f field e m i s s i o n have b e e n used for m a n y years to m e a s u r e c h a n g e s in w o r k f u n c t i o n i n d u c e d b y a d s o r p t i o n . T h e p r o b e hole t e c h n i q u e d e v e l o p e d b y Mfiller [1 ] m a d e it possible to m e a s u r e w o r k f u n c t i o n differences o n single crystal faces o f t h e e m i t t e r . This m e a s u r e m e n t p r o c e d u r e is b a s e d o n t h e original c a l c u l a t i o n b y F o w l e r a n d N o r d h e i m [2] o f t h e c u r r e n t versus field c h a r a c t e r i s t i c s o f a free-electron S o m m e r f e l d t y p e e m i t t e r . T h e i r result is k n o w n as t h e F o w l e r - N o r d h e i m ( F N ) e q u a t i o n for t h e c u r r e n t d e n s i t y J as a f u n c t i o n o f t h e field F a n d w o r k f u n c t i o n 9:
J = [AF2/c~t2(y)] exp [-c(~3/2u(y)/F], where
(la)
T.V. Vorburger et al./FieM emission work ]unctions
418
y = (e3F)l/2/~b,
(lb)
c = ~ (2m) 1/2/eh,
( 1c)
andA is a constant, u(y) and tO') are slowly varying functions tabulated by Miller [3]. t(y) is related to v(y) by the equation tO') = u o ' ) - - ~ y dv/dy. In general, the field at the emitter surface cannot be measured, only the voltage V between the emitter and the anode, so it is convenient to write F = ~V.
(2)
With this expression we define the slope of In (J/V 2) with respect to 1/V as the slope of the FN equation: SFN =
a In ( J / V 2) _ O(1/V) - -cdp3/2so')/~'
(3a)
where
s(y) = v(y) - ~y dv/dy.
(3b)
The change in work function due to adsorption can be found by taking the ratios of the slopes of the FN plots before and after adsorption, S FN O and SIN respectively
i,~o ~2/3 " (~bl/~b0) = (.~,1 "~ FN'~FN'
(4)
This technique has been very profitably applied to adsorption studies especially by Gomer and co-workers (for example, see Engel and Gomer [4] ). In order to use eq. (3) to obtain absolute work functions or relative work functions for various crystal planes,/3 must be determined. Since the emitter is not spherical in shape,/3 will vary across the surface of an emitter [5] and will depend upon the size of the crystal plane developed on the emitter [6]. An illustration of how serious this effect may be is furnished by the work of Todd and Rhodin [7] who used the assumption that/3 was uniform over the emitter to measure the work function of (110)W for a thermally annealed field emitter and a field-evaporated emitter. The resulting apparent work functions of 5.9 and 5.4 V for the two cases, respectively, indicated that the end form o f the emitter was changing. The (110) plane was much larger in the thermally annealed case, causing a reduction in ~. Young and Mfiller [8] realized this problem in their 1962 paper and suggested three techniques to independently measure the field. One of their proposed techniques was to use the energy dependence of the field emitted electrons. Young and Clark [9] elaborated on this procedure in their 1966 papers. The concept was based on the hypothesis that the energy distribution from a clean crystal face can be represented by a free electron energy distributionj~, which has the form [6,10]
1.;@) = jo d - 1 ee/d f(e) '
(Sa)
where J0 is the total current density, e is equal to the energy of the electron relative to the Fermi energy EF, f(e) is the Fermi function, and
T.V. Vorburger et al./Field emission work functions
419
d-1 = 2(2m/h2)l/2 ¢1/2 t(y)/eF.
(5b) .t
Eq. (5a) is an approximate form of the more exact expression for/0 to be shown as eq. (8b). Young and Clark [9] used eq. (5a) to show that the slope of the logarithm of the energy distribution with respect to energy is SED = cqln/0(e)/0e = d - l ,
(6)
which when combined with eq. (3) yields an expression for ¢ in terms of the measurable quantities SFN and SED : q~e= -- ~SFN tCv)/[VSED s(y)].
(7)
We use the term ~be to represent experimental values for the work function obtained by applying eq. (7). Although this relationship has been used by several workers to measure an absolute work function, the resulting values of work functions vary quite markedly for the same crystal face [11,12] and usually give values which differ considerably from those obtained by other methods [13,14]. These observed discrepancies in q~e are a consequence of the non-free electron nature of the energy distribution. This effect has been studied by Swanson and Crouser [11] and by Hummer and Bell [15] for several single crystal planes of tungsten. The latter authors defined an enhancement factor R(e) by
]'(e) = R(e)l 0 (e),
(8a)
where
1"o(e) = (K/d) exp [ - c ( ~ - e ) 3/2 o(y')/F] f(e),
(8b)
withy' equal to (e3F)l/2/(¢- e) and K, a constant. In eqs. (8),/'~(e) is the calculated free electron energy distribution [6] as a function of field and work function and ]'(e) is the measured total energy distribution. Ifj'(e) were free electron like then R(e) would be a constant. However, Plummer and Bell's results (fig. 1) for several clean faces of tungsten indicate that there is a great deal of structure in the R(e) curves. In terms of eq. (8) the slope of the logarithm of the energy distribution is written as .W
'-~o-
lnj'_OlnR ae
ae
Olnl0 ~ a~'
SE D - a lnR(e) + ~c(¢ - e) 1/2 t(y')/Y.
(9)
(10) ae Using eqs. (3) and (10) we have the equivalent relationship to Young and Clark's [9] [eq. (7)] for a non-free electron metal,
72 V. Vorburger et al./P)eM emission work functions
420
(112)
-2
. ,,r'--
I
I
-1 5
10
~ "" ~',~
I
-0.5
lO R
I
0
E - Ef(eV)
Fig. l. The enhancement factor curves R(e) for the four low index planes of tungsten. The freeelectron energy distribution j'o(e) was calculated using the following work functions: ~ (110) = 5.25 V, ~ (i 00) = 4.64 V, 4~(112) = 4.90 V, and q~(111) = 4.45 V. The dashed curves indicate the change in the structure for the (111) and (112) faces as the electric field was changed. The approximate electric fields were: 0.27 V/A for the dashed (111) curve, 0.33 V/A for the solid (111 ) curve, 0.29 V/A for the dashed (112) curve, 0.38 V/A for the solid (112) curve, 0.37 V/A for the solid (100) curve, 0.33 V/A for the solid (110) curve. Eqs. (9), (10), and (11) are derived for electron energies greater than 0.1 eV below the Fermi energy so that f(e) = 1 to a very good approximation. The presence o f 0 lnR(e)/Oe in eq. (11) demonstrates both the source of error in applying eq. (7) to a non-free electron metal and a reason for inconsistency in published data. If ~ lnR/Oe is nearly field independent, then the difference in the measured slope o f the logarithm of the energy distribution given in eq. (9) from the free electron distribution will be a constant. This implies that the error introduced by the non-free electron term will be less important as the field is reduced since SED is proportional to 1/V, i.e., the band structure effects are less important at low fields. The value o f ~e obtained using eq. (7) is therefore field dependent. Experimentally the values o f 0 lnR/Oe for single crystal planes o f tungsten can vary by ~ ( - 5 to +5) eV -1 while SED falls in the range (5 to 12) eV -1 depending upon the field and work function. Therefore it is apparent that the effect of the non-free
T.V. Vorburger et al./Field emission work func=,t,,s
421
VOLTAGE
~ - - - ~
i
i
t
i 3~°° ,
i
i
i nsl°° i
i
,
W(LOO) T=295K
t-
-2
\
-~=-.15eV
Z U.
Tt_ "
~---I~:- 5 5 e V
U')
c=-.15eV o°1 m
A
I
I
In
.225
o
.250
ELECTRIC
0 bJ
FIELD
.3OO
°275
(V/~)
(I) I
J
rid
VOLTAGE
(kl
I
I!
~l
,
I 3510 I
I
I w
o.5/
I
I 4/°O '
I
4500
I
I
I
(112)
T =295 K ,e = - . 5 5 e V
T.
;_-T~
,.e,
i •
_=
~:-.35eV
LIr=-.35eV
0.1
O
%'-(=-.55
e : - . 15eV
eV
.250
ELECTRIC
.275
FIELD
.3OO
.325
(V/~)
Fig. 2. Values of Ce 1 [calculated by eq. (7)] versus electric field for W (100) and W (112) at three different values of electron energy. For all of the figs. 2 through 4, each data point is an average of measurements taken at 0.01 eV intervals over a 0.1 eV energy range centered about the energy shown. Straight lines are fitted to the data points. As shown by eq. (11), the slope of each line is proportional to a lnR/ae and the zero field intercept (ta) is equal to ( ¢ 1 - e ) 1/2 ¢i -3/2 . Each error bar represents one standard deviation of random error.
T.V. Vorburger et al./FieM emission work functions
422
electron contribution cannot be ignored, at least for tungsten. In fig. 2 we illustrate the band structure effect. The inverse o f the work function ~bel obtained from applying eq. (7) to the data for the (100) and (112) crystal faces is plotted as a function o f field. The three values of e correspond to regions where l n R / 0 e is positive, negative, and nearly zero. Ce is a function o f b o t h e and F for a given face, and in fact the values range from 3.8 eV to 6.0 eV for (112) W and from 3.0 eV to 10.0 eV for (100) W. This is a consequence of the non-free electron behavior of the energy distributions or in terms of eq. (11) it means that 0 lnR/Oe is energy dependent and appreciable compared to the measured SED. Consequently, the method o f u s i n g eq. (7) at one field and one energy cannot be validly applied to obtain absolute work functions. The objective of this paper is to develop a technique to eliminate the contribution from the 8 lnR/Oe term. Since SED V is a constant in the free electron model the obvious way to eliminate the contribution of a lnR/Oe in eq. (11) would be to extrapolate to zero field. The condition under which tiffs extrapolation is justified is the subject o f the next section.
2. Reformulation of field emission tunneling Penn and Plummer have shown that the field emission energy distribution j'(e) from a metal can be written in the form [16]
j'(e) = ( 2 h / m ) f ( e ) ~ ( N m ) 2 D 2 ( e m - e T ) 8 ( e r a - e ) ,
(12)
/'?l
where N m is determined by the normalization of the metal wave function, D 2 is the usual barrier penetration probability, em is the total energy o f an electron in the solid, and e T is the transverse energy. The equation was calculated using the transfer Hamiltonian approximation. We want to show how eq. (12) can be written in the form of eq. (8a) so that we may use eq. (11) to obtain work functions. The wave function normalization term N m will in general be a function o f b o t h e m and e T : N 2 _ - N m2( e m ,eT).
(13)
As a first approximation to eq. (12) we may assume that N m is a function only of e m . We assume that (Nm) 2 changes slowly with respect to the exponential tunneling term D 2 as e T is varied. This approximation was justified for the (100) face by observing that the R curves calculated [6] from data using eq. (8) were independent of field to within the experimental uncertainty o f the data points [16]. This approximation allows eq. (12) to be factored into the form .t
/'(e) = R(e)10(e),
(14)
where as before l"~(e) is the free-electron total energy distribution, and R is given by eq. (32) in ref. 16:
T.V. Vorburgeret al./FieM emission work functions
R(e)
0) ] 2
Xm )
o)]
Xm ) '
(15)
-
2
423
where N O and p~ are the normalization factor and perpendicular density of states for a free electron metal. P~m is the perpendicular density of states of the metal at the classical turning point X m and Nm is evaluated at total energy e and zero transverse energy in this approximation. The above approximation yields an R factor, eq. (15), which is field independent. This fact allows us to use eq. (11) to obtain the work function ¢ by measuring experimentally the quantity ~SED(e)V/SFN for a given crystal face and energy e as a function of voltage V. This quantity is then plotted as a function of voltage and extrapolated linearly to V = 0. The intercept is ( ¢ - 6 ) 1/2 ¢-3/2. The slowly varying function t(y') can be estimated by calculating the field-voltage relationship from the slope of the FN plot using a reasonable work function. If our assumptions are correct the only error introduced in this linear extrapolation is in the estimate of t(y') which is 1.0 at F = 0 and approximately 1.04 at F = 0.3 V/A. The technique suggested here is similar to one suggested by Young and Clark [9], who were concerned with the effects of patch fields on the measured value for the work function of the (110)W face. They derived an expression for ¢ that is not sensitive to the patch effects. Their experimental method, like ours, involves measurements o f J [eq. (1)] and SED as the applied field is varied and their relationship for ¢ is d(lnJS2D ) -
2_¢s(y)
d(SED)
(16)
3 t(y)'
which is a good deal more accurate than eq. (7). Neglect of the structure factor R(e), however, introduces a small error. If we substitute the expressions for J [eq. (1)] and SED [eq. (11)] into eq. (16), we get d(lnJS2D ) d(SED)
2
¢3/2
s(y) .
2alnR/ae
- 3 (¢_e)1/2 t(y') r SED(SED_ 3 lnR/ae)"
(17)
The error involved in neglecting tile second term can be as large as 0.4 V for the (100) W face. Until the present work, there have not been sufficient data taken as a function of field to determine the validity of either eq. (11) or eq. (17).
3. ResuRs
Field emission energy distributions versus field were taken for the (100), (112), (013), and (111) faces of tungsten. For the (112) face, data were taken at 295 K and
424
T.V. Vorburger et al./Field emission work functions
at 78 K. Although, there has been a good deal of controversy over the value of for (110) W [7], we did not make measurements on this face, because the data of Plummer and Bell [15] (fig. 1) indicate that the band structure factor R(e) is relatively constant for (110)W as compared with the (100), (111) and (112) faces. Hence, the (110) face would not present a critical test of the extrapolation technique. Furthermore, the flatness of the R(e) curve between e = 0 eV and e = - 0 . 4 eV suggests that the value for q~of 5.25 V assumed by Plummet and Bell is very close to being the correct one, a result which substantially agrees with those obtained by other methods [13,14]. Conversely, the (100) face (which was measured) should be the severest test of this technique since the structure at e ~ - 0 . 3 5 eV (fig. 1) is the most pronounced of any observed for tungsten. The instrument employed for these measurements was developed by Kuyatt and Plummer and is discussed in detail elsewhere [17]. The pressure was ~ 3 X 10 11 torr as measured by a trigger discharge gauge and the time duration of a typical energy distribution was ~ 100 sec. The emission tips used ranged in size from about 1000A to about 2 5 0 0 A radius [18]. Before each distribution measurement at any given field, the tip was cleaned by resistive heating to a temperature (~ 2500 K) sufficient to produce energy distributions characteristic of the clean surface [15]. Under these conditions, the diameter of the (112) plane subtended an angle o f about 7.5 ° on the otherwise curved surface of the tip. The (100) plane subtended an angle of about 5 ° 6 °. These estimates are based on data taken after similar heating procedures by Becker [19] and Drechsler and Liepack [20]. Under these conditions, the (013) and (111) directions were not true crystal planes but rather positions of high curvature on the emitter. These directions were studied in order to determine whether valid field emission work function measurements were possible at such sites. Fig. 2 shows the plots o f ~bel (or g SED Vs(F)/[SFNt(v')] ) versus F at three different energies for the (100) and (112) faces studied at a temperature of 295 K. Each error bar represents a statistical uncertainty of one standard deviation. The results obtained by extrapolating the data to zero field are called q~l(e) and are shown in fig. 2. Table 1 shows all of the values for q~l obtained for the (100) face. The derived work function for this face of 4.57 -+ 0.14V is found by taking the weighted mean of these data. If the extrapolation technique is correct the extrapolated work function ~b1 obtained for a given face should be independent of e. This is true for the (100) and (013) data, but a closer inspection of the (111) and (112) data reveals a systematic dependence of ~1 with the values of ~ lnR/Oe. If this latter quantity is positive the work function as determined by eq. (11) is slightly higher than the mean value, and if it is negative the work function is slightly lower. This systematic deviation is shown in fig. 3 for the (112) data taken at 295 K, where q~ll(e) is plotted versus lnR/~e. The (112) data taken at 78 K reveal the same systematic variation of ~b1 with 0 lnR/Oe. Examination of eq. (11) yields two possible causes for this effect: (1) R(e) is
T.V. Vorburger et al./FieM emission work functions
425
Table 1 Results for the (100) face o f t u n s t e n (the data shown in columns II and III are average values o f data taken at 0.01 eV intervals over a 0.1 eV range centered about the initial energy shown in c o l u m n I; the range o f values o f q~e in column II illustrates the variation o f ~e with electric field) Initial energy (eV)
Ce (V)
~1 (V)
-0.15 -0.25 -0.35 -0.45 -0.55
7.79.9 8.6-12.7 4.55.2 2.73.0 3.03.7
4.88±0.23 4.48±0.19 4.18±0.21 4.79±0.47 4.69±0.62
Mean value: ~l = 4.57 ± 0.14 V.
field dependent, or (2) the tunneling barrier is not adequately described by the function t(y'). In the latter case, the variation of t(y') with energy e would have to be an order of magnitude larger than that calculated for the conventional tunneling model in order to explain the deviations observed for the (112) and (111) faces. It is unlikely that the image potential correction is so unrealistic to warrant such a large correction. Rather, the correlation between the deviations in ~1 and a lnR/ae
I
W(lO0)
I I
.26
I .24
--
I
.20
I
I
I
__.1. 7.
I
I
I
i
j
1
i
J
L______J_
I .26
--
W(ll2)
I
-
I .24
--
.20
--
.16
--
I
-5.o-,.o-3.o-2.o-,.o o aln R/a~(eV
,.o "l)
,.o
3.0
4.0
5.0
Fig. 3. ~i -t versus a lnRlae for the (100) and (112) faces o f tungsten. The values for ~ lnRl~e are calculated from t h e slopes o f the straight lines fitted to t h e data such as those shown in fig. 2. Also shown are the zero field intercepts (o) calculated by fitting the points with straight lines.
426
T.V. Vorburger et al./f'ieM emission work functions
indicate that R must be field dependent. Tiffs observation is borne out by the field dependence of the R(e) curves shown in fig. 1. As the field is increased the structure in both curves becomes more pronounced. In addition the peak in the (111) curve at " -0.75 eV is shifted in energy. By comparison, no such field dependence was observed for W (100) and W (013) within the uncertainty of the measurements. The field dependent behavior is contrary to the prediction of eq. (15), which is derived using tile assumption that the wave function normalization constant N m is not a function of the transverse energy e T. A better approximation to account for the anisotropic nature of the surface wave function would be to expand (Nm) 2 about ET = 0"
w~(~,%) = N~(~,0) + ~T aNm2(< 0)/a%"
(18)
Then from eq. (12), 2 (em, 0) ONm f ( e ) = Ro(e)Y'o(e) + (2h/m)J(e) SeTm ~eT D2(em - e T ) ~(e-em)' (19) where R0(c ) is given by eq. (15). If we transform the sum to an integral this equation becomes i'(c) = R0(c)J'0(e) + (2h/rn)f(c)
R(c) = Ro(C ) + (2h/m)
0N2(c, a e ~ 0 ) ¢o ~ax cT D2(e-cT) de T,
aN2 (e, 0) J'0T ., , OcT 10
.t
(20)
(21)
.¢
where ]0T is the tangential energy distribution, and lo(e ) is the free electron energy distribution:
~ax /0T(C) = f ( e ) J cT DZ(e--eT ) de y. 0 e max
(22a)
Jo(C) = f ( e ) f 0
(22b)
,
,T
D2(e--cT ) de T.
Since the tunneling probability D2(e) is proportional to exp(c/d) in the approximation of eq. (5a), we get .#
.t
]OT/Jo ~- d = ehF/[(8mCp)l/2t(y')] .
(22c)
Use of eq. (22c) in eq. (21) then gives R(e) = Ro(e ) + A(e) F, where
(23a)
T.V. Vorburgeret al./Fieldemissionworkfunctions
A(e) = (2m3¢)1/2 t o ' ' )
BeT
427
(23b)
Consequently,
~lnR(e)=I~--~[lnRo(e)] ]
[1 +B(e)F] + O ( F 2 ) ,
(24a)
where
OA/Oe A B(e)- ~Ro/~e RO .
(24b)
Eqs. (24) show that the slopes of the curves in fig. 3 deviate from ~ [SO')/SFNtO")] (~ lnR0/Oe ) because of the term B(e)F. An order of magnitude estimate of the varia.t tions in j'(e) with respect to lO(e) can be illustrated for a specific but typical case, (112) W. If we choose ¢(112) = 4.9 V, then/~(e) can be calculated. For a field of 0.3 V/A and an energy e = - 0 . 4 5 eV, we have that SED = O ln/'(e)/Oe = 9.4 eV -1 ,
lnR/Oe = 1.0 eV -1 , (O/OF)(~ lnR/Oe) = 2.3 (eV'V/A) -1 .
(25)
Since the data for (112)W (fig. 3) indicate that there is an error in the determination of ¢1 which is proportional to ~ lnR(e)/Oe, a quadratic extrapolation o f the data to V = 0 is called for instead of a linear extrapolation. We cannot calculate the quantity B in eq. (24) without first assuming a value for ¢, so it would be very difficult in practice to carry out a quadratic extrapolation. Instead, a linear extrapolation of the (112) results for ¢1 (fig" 3) to a value of zero for ~ lnR(e)/~e yields a fitted value for ¢, henceforth labelled ¢2, which eliminates these systematic errors in ¢1. In table 2 we list the work function obtained by the three techniques outlined in this paper: (1) several sets o f measurements [11,12] based on the one point determination Ce calculated via eq. (7); (2) the results ¢1 for the linear extrapolation of eq. (11) to zero field; and (3) the results ¢2 where further extrapolation was made to zero for a lnR/Oe. The values for Ce in column IV are the extrema of the measured values which were taken over a range of initial energies ( - 0 . 7 eV to - 0 . 1 eV) and fields (0.22 V/A to 0.32 V/A). The values of Swanson and Crouser and Lea and Gomer in columns II and III result from fitting a single straight line to the SED over an energy range between 0 and ~ - 0 . 6 eV. This technique averages away some, but not all o f the error caused by the structure in the energy distributions, and in fact cannot be used for the (100) face because the pronounced structure indicates that the SED cannot be reasonably approximated by a straight line. The final result for the (100) face indicates that a reliable value for the work function can be obtained in spite of the structure in the energy distribution. As
T.V. Vorburgeret al./FieM emission work functions
428
Table 2 Results in volts for field emission work functions of tungsten W crw'~l face
4e (Ref. 1 l )
Oe 4e (Ref. 12) (Present work)
(100)
2.7
(112)
4.84, 5.05
(013)
4.16,4.34
(111)
4.80
5.02
4.78
12.7
3.8 3.5
6.0 (295 K) 6.5 ( 78 K)
3.3
6.1
3.5 .... 5.9
Ol
42
4.57 + 0.14 4.92 + 0.18]mean = 4.82 -+ 0.27/4.89-+0.15 5.19 +- 0.16 4.83 -+ 0.38
The quoted uncertainties are equal to one standard deviation. The values for 41 in column V are mean values of 41 taken over the range of energies between ~ -0.6 eV and -0.1 eV. In the case of the (112) and (111 ) faces, the result of 41 varies systematically with the value of 0 In R~ 0e. The value of 42 in these cases represents the result obtained by extrapolating the curve for 41 versus 0 In R/Oe (as in fig. 3). Column II shows the results of Swanson and Crouser [ 11 ] for 4e. The three pairs of measurements were made under various conditions of tip orientation, flashing, and annealing. The results of Lea and Gomer are shown in column III. Column IV shows the extrema for all 4e'S measured over the range of electric fields and electron energies studied. shown in table 3 the value o f 4.57 -+ 0.14 V is consistent with the result o f Strayer et al. [13] o b t a i n e d by the field emission retarding potential m e t h o d ( F E R P ) and and with m e a s u r e m e n t s using t h e r m i o n i c emission [13,14]. In the case o f the (112) face, we are not aware o f any F E R P measurements, and there is an inconsistency b e t w e e n the two values d e t e r m i n e d by t h e r m i o n i c emission [14]. Our result o f 4.89 + 0.15 eV is consistent with a t h e r m i o n i c m e a s u r e m e n t by A z u z o v [14] and w i t h the earlier field emission results d e t e r m i n e d at a single field [ 1 1 , 1 2 ] . Tile syst e m a t i c error resulting from neglect o f 0 lnR/Oe in eq. (7) for these m e a s u r e m e n t s would not be as great as for the (100) face since the structure in the energy distribution o f the (112) face is less dramatic, and the error is to some extent averaged away as discussed in the previous paragraph. The results for the (013) face are similar to those for the (100) insofar as there is no correlation b e t w e e n ¢1 and ~ lnR/Oe. Tile results indicate a m u c h higher value for the w o r k f u n c t i o n than the values o f 4.3 V to 4.35 V that had b e e n observed previously [11,14,21 ] , but the only measurements taken on this face since 1937 (that we are aware of) all utilize field emission. Our own single field data are consistent w i t h those o f Swanson and Crouser [11]. When the results are averaged over the range o f energies b e t w e e n - 0 . 7 eV and - 0 . 1 eV below the Fermi energy (fig. 4), the derived w o r k f u n c t i o n is 4.25 V which agrees with their results o f 4.16 V and 4.34 V m o s t likely because their data were averaged over a similar energy range. The data for the (111) face reveal a systematic d e p e n d e n c e o f q51 on the value o f ~ In R~ 3e that is similar to the (112) result. In this case, however, most o f the points (4~1, ~ lnR/Oe) lie on the left side o f the origin. For this reason and because
T.V. Forburger et al./Field emission work functions
429
Table 3 Summary of tungsten work functions in volts determined by various methods (column 3 contains results for the field emission retarding potential method; column 4 contains average values from several thermionic emission experiments discussed by Rlvlere " '" in his 1969 review; column 5 gives the contact potential difference results of Hopkins and Pender also discussed by Rivi~re) W crystal face
Present work
FERP (Ref. 13)
Thermionic emission (Ref. 14)
CPD (Ref. 14)
(100)
4.57 ± 0.14
4.63 ± 0.02
4.58 ± 0.08
4.67 + 0.04
(112)
4.89 ± 0.15
(013)
5.19 -+ 0.16
4.80 ± 0.05 a 5.24 -+ 0.05
(110) (111)
4.83 -+ 0.38
5.25 ± 0.02
5.30 -+ 0.12
4.47 ± 0.02
4.40 -+ 0.03
5.05 -+ 0.02 5.15 ± 0.03
a Rivi~re did not calculate an average value for the (112) face since the two results shown are in disagreement.
of the fact that the (111) data were taken for only four different electric fields the second extrapolation yields a value for 92 which has a large statistical uncertainty.
j
I
i
I
r
I
I
6.0
5.0
"~ 4.0
s
3.C
I
-0.8
I
-0.7
I
-0.6
I
-0.5 ==E-El
I
-0,4
I
-0.3
I
-0.2
-0.1
(eV)
Fig. 4. Single field determinations Oe for W (013) versus energy. The filled circles are values for q5e measured over a 0.1 eV energy interval centered about the energy indicated. Each open circle corresponds to an average value for Oe taken over the entire range between - 0 . 1 eV and the energy indicated. The squares show the values of ¢~e measured by Swanson and Crouser for the (013) plane.
430
T.V. Vorburger et al./FieM emission work Junctions
The final result of 4.83 + 0.38 V is higher than that obtained by other methods, but this difference is not significant in view o f the large uncertainty.
4. Conclusions Previous determinations of the absolute value of the work function from field emission total current and energy distribution measurements have produced contradictory results. The determinations were based on the Fowler-Nordheim equation which relates the field emission current of a free electron metal to its work function. We have shown that "band structure" effects have a sufficiently important effect on the field emission energy distribution that they must be taken into account if correct (and consistent) values o f tile work function are to be obtained. We have taken "band structure" effects, caused by tire periodic lattice potential, into account by writing j'(e) = R(e)/'~(e) where/'(e),J'o(e ) are the field emission currents at energy e of the actual metal and o f the free electron-like metal respectively. A theoretical analysis has produced an expression for R(e). This expression shows that R depends not only on energy but for some faces has a dependence on electric field as well. The dependence of R on both energy and field must be considered in order to determine the work function, and a procedure for doing this has been given in the text. This technique was shown to give reliable absolute work functions for the (100) and (112) planes o f a W emitter. The value obtained for (013) W was much higher than what we would expect and preliminary data indicates a similar deviation exists on the (111) face. It is possible that there is an error in the high index directions, that is caused by the roughness of the surface, which is not adequately represented by our one dimensional model. Specifically, tile equation for determining 0 [eq. (11)] is based on the expression for tile free electron tunneling probability, eq. (8b), which is derived for a one-dimensional tunneling model. If this model were not valid, because of surface roughness on an atomic scale or because of the surface curvature, eq. (8b) would be incorrect. In such a case, one would expect to observe such phenomena as the shifting of the (111) structure with increasing electric fields. Alternatively, since the (111) and (013) are not well developed planes the probe hole could accept sufficient electron emission from surrounding regions (which may have different work functions and may be experiencing different electric fields) to cause some error in the results [22]. It is not felt that this edge emission is affecting the results for the (100) and (112) faces because of the large sizes of these planes and because the results agree with those obtained by other methods. It might be possible to determine whether any or all of these possible effects is significant in the high index directions by taking work function measurements for emitters with varying sizes or by varying the sizes of the planes themselves using field evaporation [22]. This technique for measuring work functions is relatively complicated, but it
T. II. Vorburger et al./FieM emission work ]unctions
431
does have the capability of measuring absolute (not relative) work functions for the low index planes which are well developed on a field emitter [(110) [15], (100), (112) for W]. We view this technique as primarily useful for field emission energy distribution studies of materials where reliable absolute work functions are not available. For example, to apply the formulation of Penn and Plummer [ 16] to measure the "surface density of states" a free electron energy distribution must be calculated first for the appropriate work function and field. The work function can be obtained for well developed planes by applying the above techniques, and the field can then be calculated from the slope of the Fowler-Nordheim plot [15].
Acknowledgements The authors are grateful to Drs. C.J. Powell, C.E. Kuyatt, and A.J. Melmed for a number of helpful discussions.
References [1] [21 [3] [4] [5]
E.W. M~ller, J. Appl. Phys. 26 (1955) 732. R.H. Fowler and L. Nordheim, Proc. Roy. Soc. (London) A 119 (1928) 173. H.C. Miller, J. Franklin Inst. 282 (1966) 382. T. Engel and R. Gomer, J. Chem. Phys. 52 (1970) 5572. L.W. Swanson and A.E. Bell, Advan. Electron. Electron Phys. 32 (1973) 193. [6] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487. [7] C.J. Todd and T.N. Rhodin, Surface Sci. 36 (1973) 353. [8] R.D. Young and E.W. M~ller, J. Appl. Phys. 33 (1962) 91. [9] R.D. Young and H.E. Clark, Phys. Rev. Letters 17 (1966) 351; R.D. Young and H.E. Clark, Appl. Phys. Letters 9 (1966) 265. [I0] R.D. Young, Phys. Rev. 113 (1959) 110. [11] L.W. Swanson and L.C. Crouser, Phys. Rev. 163 (1967) 622. [12] C. Lea and R. Gomer, J. Chem. Phys. 54 (1971) 3349. [13] R.W. Strayer, W. Mackie and L.W. Swanson, Surface Sci. 34 (1973) 225. [14] J.C. Rivi~re, in: Solid State Surface Science, Vol. 1, Ed. M. Green (Dekker, New York, 1969) p. 179. [15] E.W. Plummer and A.E. Bell, J. Vacuum Sci. Technol. 9 (1972) 583. [16] D. Penn and E.W. Plummer, Phys. Rev. B9 (1974) 1216. [17] C.E. Kuyatt and E.W. Plummer, Rev. Sci. Instr. 43 (1972) 108. [18 ] Estimates of the emitter radius were based on considerations given by R.H. Good and E.W. M~ller, in: Handbuch der Physik, Vol. 21, Ed. S. Fl~gge (Springer, Berlin, 1956) p. 176. [191 J.A. Becker, Bell Syst. Tech. J. 30 (1951) 907. [20] M. Drechsler and H. Liepack, in: Adsorption et Croissance Cristalline (CNRS, Paris, 1965) p. 49. [21 ] C.E. Mendenhall and C.F. de Voe, Phys. Rev. 51 (1937) 346. [22] A.J. Melmed, 20th Field Emission Symposium, Pennsylvania State University, University Park, Pennsylvania, August 1973 (unpublished). This author has measured the slope of Fowler-Nordheim plots for W (110) and Ru (1010) as a function of plane size and observed variations in the derived work function of several percent.
FIELD EMISSION WORK FUNCTIONS T.V. V O R B U R G E R a n d D. P E N N
National Bureau of Standards, Washington, D.C. 20234, U.S.A. and E.W. P L U M M E R
Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A. Received 16 July 1974; manuscript received in final form 25 October 1974
Field emission has proven to be a very useful technique for obtaining work function changes from single crystal planes or from the whole emitter. The inability to independently measure the electric field has limited the accuracy of field emission total current measurements for determining absolute work functions. Young and Clark's method of combining field emission energy distribution measurements and total current versus voltage measurements to eliminate the electric field as an unknown is not adequate because it does not take into account the effects of the surface density of states present in the energy distribution. In this paper we discuss a technique to overcome this problem, which involves a series of measurements taken as a function of electric field and an extrapolation to zero field. The method yields reliable values of the work function for the low index (fiat) (100) and (112) planes of tungsten but not for the high index (rough) (013) and (111) faces.
I. i n t r o d u c t i o n T h e c u r r e n t - v o l t a g e characteristics o f field e m i s s i o n have b e e n used for m a n y years to m e a s u r e c h a n g e s in w o r k f u n c t i o n i n d u c e d b y a d s o r p t i o n . T h e p r o b e hole t e c h n i q u e d e v e l o p e d b y Mfiller [1 ] m a d e it possible to m e a s u r e w o r k f u n c t i o n differences o n single crystal faces o f t h e e m i t t e r . This m e a s u r e m e n t p r o c e d u r e is b a s e d o n t h e original c a l c u l a t i o n b y F o w l e r a n d N o r d h e i m [2] o f t h e c u r r e n t versus field c h a r a c t e r i s t i c s o f a free-electron S o m m e r f e l d t y p e e m i t t e r . T h e i r result is k n o w n as t h e F o w l e r - N o r d h e i m ( F N ) e q u a t i o n for t h e c u r r e n t d e n s i t y J as a f u n c t i o n o f t h e field F a n d w o r k f u n c t i o n 9:
J = [AF2/c~t2(y)] exp [-c(~3/2u(y)/F], where
(la)
T.V. Vorburger et al./FieM emission work ]unctions
418
y = (e3F)l/2/~b,
(lb)
c = ~ (2m) 1/2/eh,
( 1c)
andA is a constant, u(y) and tO') are slowly varying functions tabulated by Miller [3]. t(y) is related to v(y) by the equation tO') = u o ' ) - - ~ y dv/dy. In general, the field at the emitter surface cannot be measured, only the voltage V between the emitter and the anode, so it is convenient to write F = ~V.
(2)
With this expression we define the slope of In (J/V 2) with respect to 1/V as the slope of the FN equation: SFN =
a In ( J / V 2) _ O(1/V) - -cdp3/2so')/~'
(3a)
where
s(y) = v(y) - ~y dv/dy.
(3b)
The change in work function due to adsorption can be found by taking the ratios of the slopes of the FN plots before and after adsorption, S FN O and SIN respectively
i,~o ~2/3 " (~bl/~b0) = (.~,1 "~ FN'~FN'
(4)
This technique has been very profitably applied to adsorption studies especially by Gomer and co-workers (for example, see Engel and Gomer [4] ). In order to use eq. (3) to obtain absolute work functions or relative work functions for various crystal planes,/3 must be determined. Since the emitter is not spherical in shape,/3 will vary across the surface of an emitter [5] and will depend upon the size of the crystal plane developed on the emitter [6]. An illustration of how serious this effect may be is furnished by the work of Todd and Rhodin [7] who used the assumption that/3 was uniform over the emitter to measure the work function of (110)W for a thermally annealed field emitter and a field-evaporated emitter. The resulting apparent work functions of 5.9 and 5.4 V for the two cases, respectively, indicated that the end form o f the emitter was changing. The (110) plane was much larger in the thermally annealed case, causing a reduction in ~. Young and Mfiller [8] realized this problem in their 1962 paper and suggested three techniques to independently measure the field. One of their proposed techniques was to use the energy dependence of the field emitted electrons. Young and Clark [9] elaborated on this procedure in their 1966 papers. The concept was based on the hypothesis that the energy distribution from a clean crystal face can be represented by a free electron energy distributionj~, which has the form [6,10]
1.;@) = jo d - 1 ee/d f(e) '
(Sa)
where J0 is the total current density, e is equal to the energy of the electron relative to the Fermi energy EF, f(e) is the Fermi function, and
T.V. Vorburger et al./Field emission work functions
419
d-1 = 2(2m/h2)l/2 ¢1/2 t(y)/eF.
(5b) .t
Eq. (5a) is an approximate form of the more exact expression for/0 to be shown as eq. (8b). Young and Clark [9] used eq. (5a) to show that the slope of the logarithm of the energy distribution with respect to energy is SED = cqln/0(e)/0e = d - l ,
(6)
which when combined with eq. (3) yields an expression for ¢ in terms of the measurable quantities SFN and SED : q~e= -- ~SFN tCv)/[VSED s(y)].
(7)
We use the term ~be to represent experimental values for the work function obtained by applying eq. (7). Although this relationship has been used by several workers to measure an absolute work function, the resulting values of work functions vary quite markedly for the same crystal face [11,12] and usually give values which differ considerably from those obtained by other methods [13,14]. These observed discrepancies in q~e are a consequence of the non-free electron nature of the energy distribution. This effect has been studied by Swanson and Crouser [11] and by Hummer and Bell [15] for several single crystal planes of tungsten. The latter authors defined an enhancement factor R(e) by
]'(e) = R(e)l 0 (e),
(8a)
where
1"o(e) = (K/d) exp [ - c ( ~ - e ) 3/2 o(y')/F] f(e),
(8b)
withy' equal to (e3F)l/2/(¢- e) and K, a constant. In eqs. (8),/'~(e) is the calculated free electron energy distribution [6] as a function of field and work function and ]'(e) is the measured total energy distribution. Ifj'(e) were free electron like then R(e) would be a constant. However, Plummer and Bell's results (fig. 1) for several clean faces of tungsten indicate that there is a great deal of structure in the R(e) curves. In terms of eq. (8) the slope of the logarithm of the energy distribution is written as .W
'-~o-
lnj'_OlnR ae
ae
Olnl0 ~ a~'
SE D - a lnR(e) + ~c(¢ - e) 1/2 t(y')/Y.
(9)
(10) ae Using eqs. (3) and (10) we have the equivalent relationship to Young and Clark's [9] [eq. (7)] for a non-free electron metal,
72 V. Vorburger et al./P)eM emission work functions
420
(112)
-2
. ,,r'--
I
I
-1 5
10
~ "" ~',~
I
-0.5
lO R
I
0
E - Ef(eV)
Fig. l. The enhancement factor curves R(e) for the four low index planes of tungsten. The freeelectron energy distribution j'o(e) was calculated using the following work functions: ~ (110) = 5.25 V, ~ (i 00) = 4.64 V, 4~(112) = 4.90 V, and q~(111) = 4.45 V. The dashed curves indicate the change in the structure for the (111) and (112) faces as the electric field was changed. The approximate electric fields were: 0.27 V/A for the dashed (111) curve, 0.33 V/A for the solid (111 ) curve, 0.29 V/A for the dashed (112) curve, 0.38 V/A for the solid (112) curve, 0.37 V/A for the solid (100) curve, 0.33 V/A for the solid (110) curve. Eqs. (9), (10), and (11) are derived for electron energies greater than 0.1 eV below the Fermi energy so that f(e) = 1 to a very good approximation. The presence o f 0 lnR(e)/Oe in eq. (11) demonstrates both the source of error in applying eq. (7) to a non-free electron metal and a reason for inconsistency in published data. If ~ lnR/Oe is nearly field independent, then the difference in the measured slope o f the logarithm of the energy distribution given in eq. (9) from the free electron distribution will be a constant. This implies that the error introduced by the non-free electron term will be less important as the field is reduced since SED is proportional to 1/V, i.e., the band structure effects are less important at low fields. The value o f ~e obtained using eq. (7) is therefore field dependent. Experimentally the values o f 0 lnR/Oe for single crystal planes o f tungsten can vary by ~ ( - 5 to +5) eV -1 while SED falls in the range (5 to 12) eV -1 depending upon the field and work function. Therefore it is apparent that the effect of the non-free
T.V. Vorburger et al./Field emission work func=,t,,s
421
VOLTAGE
~ - - - ~
i
i
t
i 3~°° ,
i
i
i nsl°° i
i
,
W(LOO) T=295K
t-
-2
\
-~=-.15eV
Z U.
Tt_ "
~---I~:- 5 5 e V
U')
c=-.15eV o°1 m
A
I
I
In
.225
o
.250
ELECTRIC
0 bJ
FIELD
.3OO
°275
(V/~)
(I) I
J
rid
VOLTAGE
(kl
I
I!
~l
,
I 3510 I
I
I w
o.5/
I
I 4/°O '
I
4500
I
I
I
(112)
T =295 K ,e = - . 5 5 e V
T.
;_-T~
,.e,
i •
_=
~:-.35eV
LIr=-.35eV
0.1
O
%'-(=-.55
e : - . 15eV
eV
.250
ELECTRIC
.275
FIELD
.3OO
.325
(V/~)
Fig. 2. Values of Ce 1 [calculated by eq. (7)] versus electric field for W (100) and W (112) at three different values of electron energy. For all of the figs. 2 through 4, each data point is an average of measurements taken at 0.01 eV intervals over a 0.1 eV energy range centered about the energy shown. Straight lines are fitted to the data points. As shown by eq. (11), the slope of each line is proportional to a lnR/ae and the zero field intercept (ta) is equal to ( ¢ 1 - e ) 1/2 ¢i -3/2 . Each error bar represents one standard deviation of random error.
T.V. Vorburger et al./FieM emission work functions
422
electron contribution cannot be ignored, at least for tungsten. In fig. 2 we illustrate the band structure effect. The inverse o f the work function ~bel obtained from applying eq. (7) to the data for the (100) and (112) crystal faces is plotted as a function o f field. The three values of e correspond to regions where l n R / 0 e is positive, negative, and nearly zero. Ce is a function o f b o t h e and F for a given face, and in fact the values range from 3.8 eV to 6.0 eV for (112) W and from 3.0 eV to 10.0 eV for (100) W. This is a consequence of the non-free electron behavior of the energy distributions or in terms of eq. (11) it means that 0 lnR/Oe is energy dependent and appreciable compared to the measured SED. Consequently, the method o f u s i n g eq. (7) at one field and one energy cannot be validly applied to obtain absolute work functions. The objective of this paper is to develop a technique to eliminate the contribution from the 8 lnR/Oe term. Since SED V is a constant in the free electron model the obvious way to eliminate the contribution of a lnR/Oe in eq. (11) would be to extrapolate to zero field. The condition under which tiffs extrapolation is justified is the subject o f the next section.
2. Reformulation of field emission tunneling Penn and Plummer have shown that the field emission energy distribution j'(e) from a metal can be written in the form [16]
j'(e) = ( 2 h / m ) f ( e ) ~ ( N m ) 2 D 2 ( e m - e T ) 8 ( e r a - e ) ,
(12)
/'?l
where N m is determined by the normalization of the metal wave function, D 2 is the usual barrier penetration probability, em is the total energy o f an electron in the solid, and e T is the transverse energy. The equation was calculated using the transfer Hamiltonian approximation. We want to show how eq. (12) can be written in the form of eq. (8a) so that we may use eq. (11) to obtain work functions. The wave function normalization term N m will in general be a function o f b o t h e m and e T : N 2 _ - N m2( e m ,eT).
(13)
As a first approximation to eq. (12) we may assume that N m is a function only of e m . We assume that (Nm) 2 changes slowly with respect to the exponential tunneling term D 2 as e T is varied. This approximation was justified for the (100) face by observing that the R curves calculated [6] from data using eq. (8) were independent of field to within the experimental uncertainty o f the data points [16]. This approximation allows eq. (12) to be factored into the form .t
/'(e) = R(e)10(e),
(14)
where as before l"~(e) is the free-electron total energy distribution, and R is given by eq. (32) in ref. 16:
T.V. Vorburgeret al./FieM emission work functions
R(e)
0) ] 2
Xm )
o)]
Xm ) '
(15)
-
2
423
where N O and p~ are the normalization factor and perpendicular density of states for a free electron metal. P~m is the perpendicular density of states of the metal at the classical turning point X m and Nm is evaluated at total energy e and zero transverse energy in this approximation. The above approximation yields an R factor, eq. (15), which is field independent. This fact allows us to use eq. (11) to obtain the work function ¢ by measuring experimentally the quantity ~SED(e)V/SFN for a given crystal face and energy e as a function of voltage V. This quantity is then plotted as a function of voltage and extrapolated linearly to V = 0. The intercept is ( ¢ - 6 ) 1/2 ¢-3/2. The slowly varying function t(y') can be estimated by calculating the field-voltage relationship from the slope of the FN plot using a reasonable work function. If our assumptions are correct the only error introduced in this linear extrapolation is in the estimate of t(y') which is 1.0 at F = 0 and approximately 1.04 at F = 0.3 V/A. The technique suggested here is similar to one suggested by Young and Clark [9], who were concerned with the effects of patch fields on the measured value for the work function of the (110)W face. They derived an expression for ¢ that is not sensitive to the patch effects. Their experimental method, like ours, involves measurements o f J [eq. (1)] and SED as the applied field is varied and their relationship for ¢ is d(lnJS2D ) -
2_¢s(y)
d(SED)
(16)
3 t(y)'
which is a good deal more accurate than eq. (7). Neglect of the structure factor R(e), however, introduces a small error. If we substitute the expressions for J [eq. (1)] and SED [eq. (11)] into eq. (16), we get d(lnJS2D ) d(SED)
2
¢3/2
s(y) .
2alnR/ae
- 3 (¢_e)1/2 t(y') r SED(SED_ 3 lnR/ae)"
(17)
The error involved in neglecting tile second term can be as large as 0.4 V for the (100) W face. Until the present work, there have not been sufficient data taken as a function of field to determine the validity of either eq. (11) or eq. (17).
3. ResuRs
Field emission energy distributions versus field were taken for the (100), (112), (013), and (111) faces of tungsten. For the (112) face, data were taken at 295 K and
424
T.V. Vorburger et al./Field emission work functions
at 78 K. Although, there has been a good deal of controversy over the value of for (110) W [7], we did not make measurements on this face, because the data of Plummer and Bell [15] (fig. 1) indicate that the band structure factor R(e) is relatively constant for (110)W as compared with the (100), (111) and (112) faces. Hence, the (110) face would not present a critical test of the extrapolation technique. Furthermore, the flatness of the R(e) curve between e = 0 eV and e = - 0 . 4 eV suggests that the value for q~of 5.25 V assumed by Plummet and Bell is very close to being the correct one, a result which substantially agrees with those obtained by other methods [13,14]. Conversely, the (100) face (which was measured) should be the severest test of this technique since the structure at e ~ - 0 . 3 5 eV (fig. 1) is the most pronounced of any observed for tungsten. The instrument employed for these measurements was developed by Kuyatt and Plummer and is discussed in detail elsewhere [17]. The pressure was ~ 3 X 10 11 torr as measured by a trigger discharge gauge and the time duration of a typical energy distribution was ~ 100 sec. The emission tips used ranged in size from about 1000A to about 2 5 0 0 A radius [18]. Before each distribution measurement at any given field, the tip was cleaned by resistive heating to a temperature (~ 2500 K) sufficient to produce energy distributions characteristic of the clean surface [15]. Under these conditions, the diameter of the (112) plane subtended an angle o f about 7.5 ° on the otherwise curved surface of the tip. The (100) plane subtended an angle of about 5 ° 6 °. These estimates are based on data taken after similar heating procedures by Becker [19] and Drechsler and Liepack [20]. Under these conditions, the (013) and (111) directions were not true crystal planes but rather positions of high curvature on the emitter. These directions were studied in order to determine whether valid field emission work function measurements were possible at such sites. Fig. 2 shows the plots o f ~bel (or g SED Vs(F)/[SFNt(v')] ) versus F at three different energies for the (100) and (112) faces studied at a temperature of 295 K. Each error bar represents a statistical uncertainty of one standard deviation. The results obtained by extrapolating the data to zero field are called q~l(e) and are shown in fig. 2. Table 1 shows all of the values for q~l obtained for the (100) face. The derived work function for this face of 4.57 -+ 0.14V is found by taking the weighted mean of these data. If the extrapolation technique is correct the extrapolated work function ~b1 obtained for a given face should be independent of e. This is true for the (100) and (013) data, but a closer inspection of the (111) and (112) data reveals a systematic dependence of ~1 with the values of ~ lnR/Oe. If this latter quantity is positive the work function as determined by eq. (11) is slightly higher than the mean value, and if it is negative the work function is slightly lower. This systematic deviation is shown in fig. 3 for the (112) data taken at 295 K, where q~ll(e) is plotted versus lnR/~e. The (112) data taken at 78 K reveal the same systematic variation of ~b1 with 0 lnR/Oe. Examination of eq. (11) yields two possible causes for this effect: (1) R(e) is
T.V. Vorburger et al./FieM emission work functions
425
Table 1 Results for the (100) face o f t u n s t e n (the data shown in columns II and III are average values o f data taken at 0.01 eV intervals over a 0.1 eV range centered about the initial energy shown in c o l u m n I; the range o f values o f q~e in column II illustrates the variation o f ~e with electric field) Initial energy (eV)
Ce (V)
~1 (V)
-0.15 -0.25 -0.35 -0.45 -0.55
7.79.9 8.6-12.7 4.55.2 2.73.0 3.03.7
4.88±0.23 4.48±0.19 4.18±0.21 4.79±0.47 4.69±0.62
Mean value: ~l = 4.57 ± 0.14 V.
field dependent, or (2) the tunneling barrier is not adequately described by the function t(y'). In the latter case, the variation of t(y') with energy e would have to be an order of magnitude larger than that calculated for the conventional tunneling model in order to explain the deviations observed for the (112) and (111) faces. It is unlikely that the image potential correction is so unrealistic to warrant such a large correction. Rather, the correlation between the deviations in ~1 and a lnR/ae
I
W(lO0)
I I
.26
I .24
--
I
.20
I
I
I
__.1. 7.
I
I
I
i
j
1
i
J
L______J_
I .26
--
W(ll2)
I
-
I .24
--
.20
--
.16
--
I
-5.o-,.o-3.o-2.o-,.o o aln R/a~(eV
,.o "l)
,.o
3.0
4.0
5.0
Fig. 3. ~i -t versus a lnRlae for the (100) and (112) faces o f tungsten. The values for ~ lnRl~e are calculated from t h e slopes o f the straight lines fitted to t h e data such as those shown in fig. 2. Also shown are the zero field intercepts (o) calculated by fitting the points with straight lines.
426
T.V. Vorburger et al./f'ieM emission work functions
indicate that R must be field dependent. Tiffs observation is borne out by the field dependence of the R(e) curves shown in fig. 1. As the field is increased the structure in both curves becomes more pronounced. In addition the peak in the (111) curve at " -0.75 eV is shifted in energy. By comparison, no such field dependence was observed for W (100) and W (013) within the uncertainty of the measurements. The field dependent behavior is contrary to the prediction of eq. (15), which is derived using tile assumption that the wave function normalization constant N m is not a function of the transverse energy e T. A better approximation to account for the anisotropic nature of the surface wave function would be to expand (Nm) 2 about ET = 0"
w~(~,%) = N~(~,0) + ~T aNm2(< 0)/a%"
(18)
Then from eq. (12), 2 (em, 0) ONm f ( e ) = Ro(e)Y'o(e) + (2h/m)J(e) SeTm ~eT D2(em - e T ) ~(e-em)' (19) where R0(c ) is given by eq. (15). If we transform the sum to an integral this equation becomes i'(c) = R0(c)J'0(e) + (2h/rn)f(c)
R(c) = Ro(C ) + (2h/m)
0N2(c, a e ~ 0 ) ¢o ~ax cT D2(e-cT) de T,
aN2 (e, 0) J'0T ., , OcT 10
.t
(20)
(21)
.¢
where ]0T is the tangential energy distribution, and lo(e ) is the free electron energy distribution:
~ax /0T(C) = f ( e ) J cT DZ(e--eT ) de y. 0 e max
(22a)
Jo(C) = f ( e ) f 0
(22b)
,
,T
D2(e--cT ) de T.
Since the tunneling probability D2(e) is proportional to exp(c/d) in the approximation of eq. (5a), we get .#
.t
]OT/Jo ~- d = ehF/[(8mCp)l/2t(y')] .
(22c)
Use of eq. (22c) in eq. (21) then gives R(e) = Ro(e ) + A(e) F, where
(23a)
T.V. Vorburgeret al./Fieldemissionworkfunctions
A(e) = (2m3¢)1/2 t o ' ' )
BeT
427
(23b)
Consequently,
~lnR(e)=I~--~[lnRo(e)] ]
[1 +B(e)F] + O ( F 2 ) ,
(24a)
where
OA/Oe A B(e)- ~Ro/~e RO .
(24b)
Eqs. (24) show that the slopes of the curves in fig. 3 deviate from ~ [SO')/SFNtO")] (~ lnR0/Oe ) because of the term B(e)F. An order of magnitude estimate of the varia.t tions in j'(e) with respect to lO(e) can be illustrated for a specific but typical case, (112) W. If we choose ¢(112) = 4.9 V, then/~(e) can be calculated. For a field of 0.3 V/A and an energy e = - 0 . 4 5 eV, we have that SED = O ln/'(e)/Oe = 9.4 eV -1 ,
lnR/Oe = 1.0 eV -1 , (O/OF)(~ lnR/Oe) = 2.3 (eV'V/A) -1 .
(25)
Since the data for (112)W (fig. 3) indicate that there is an error in the determination of ¢1 which is proportional to ~ lnR(e)/Oe, a quadratic extrapolation o f the data to V = 0 is called for instead of a linear extrapolation. We cannot calculate the quantity B in eq. (24) without first assuming a value for ¢, so it would be very difficult in practice to carry out a quadratic extrapolation. Instead, a linear extrapolation of the (112) results for ¢1 (fig" 3) to a value of zero for ~ lnR(e)/~e yields a fitted value for ¢, henceforth labelled ¢2, which eliminates these systematic errors in ¢1. In table 2 we list the work function obtained by the three techniques outlined in this paper: (1) several sets o f measurements [11,12] based on the one point determination Ce calculated via eq. (7); (2) the results ¢1 for the linear extrapolation of eq. (11) to zero field; and (3) the results ¢2 where further extrapolation was made to zero for a lnR/Oe. The values for Ce in column IV are the extrema of the measured values which were taken over a range of initial energies ( - 0 . 7 eV to - 0 . 1 eV) and fields (0.22 V/A to 0.32 V/A). The values of Swanson and Crouser and Lea and Gomer in columns II and III result from fitting a single straight line to the SED over an energy range between 0 and ~ - 0 . 6 eV. This technique averages away some, but not all o f the error caused by the structure in the energy distributions, and in fact cannot be used for the (100) face because the pronounced structure indicates that the SED cannot be reasonably approximated by a straight line. The final result for the (100) face indicates that a reliable value for the work function can be obtained in spite of the structure in the energy distribution. As
T.V. Vorburgeret al./FieM emission work functions
428
Table 2 Results in volts for field emission work functions of tungsten W crw'~l face
4e (Ref. 1 l )
Oe 4e (Ref. 12) (Present work)
(100)
2.7
(112)
4.84, 5.05
(013)
4.16,4.34
(111)
4.80
5.02
4.78
12.7
3.8 3.5
6.0 (295 K) 6.5 ( 78 K)
3.3
6.1
3.5 .... 5.9
Ol
42
4.57 + 0.14 4.92 + 0.18]mean = 4.82 -+ 0.27/4.89-+0.15 5.19 +- 0.16 4.83 -+ 0.38
The quoted uncertainties are equal to one standard deviation. The values for 41 in column V are mean values of 41 taken over the range of energies between ~ -0.6 eV and -0.1 eV. In the case of the (112) and (111 ) faces, the result of 41 varies systematically with the value of 0 In R~ 0e. The value of 42 in these cases represents the result obtained by extrapolating the curve for 41 versus 0 In R/Oe (as in fig. 3). Column II shows the results of Swanson and Crouser [ 11 ] for 4e. The three pairs of measurements were made under various conditions of tip orientation, flashing, and annealing. The results of Lea and Gomer are shown in column III. Column IV shows the extrema for all 4e'S measured over the range of electric fields and electron energies studied. shown in table 3 the value o f 4.57 -+ 0.14 V is consistent with the result o f Strayer et al. [13] o b t a i n e d by the field emission retarding potential m e t h o d ( F E R P ) and and with m e a s u r e m e n t s using t h e r m i o n i c emission [13,14]. In the case o f the (112) face, we are not aware o f any F E R P measurements, and there is an inconsistency b e t w e e n the two values d e t e r m i n e d by t h e r m i o n i c emission [14]. Our result o f 4.89 + 0.15 eV is consistent with a t h e r m i o n i c m e a s u r e m e n t by A z u z o v [14] and w i t h the earlier field emission results d e t e r m i n e d at a single field [ 1 1 , 1 2 ] . Tile syst e m a t i c error resulting from neglect o f 0 lnR/Oe in eq. (7) for these m e a s u r e m e n t s would not be as great as for the (100) face since the structure in the energy distribution o f the (112) face is less dramatic, and the error is to some extent averaged away as discussed in the previous paragraph. The results for the (013) face are similar to those for the (100) insofar as there is no correlation b e t w e e n ¢1 and ~ lnR/Oe. Tile results indicate a m u c h higher value for the w o r k f u n c t i o n than the values o f 4.3 V to 4.35 V that had b e e n observed previously [11,14,21 ] , but the only measurements taken on this face since 1937 (that we are aware of) all utilize field emission. Our own single field data are consistent w i t h those o f Swanson and Crouser [11]. When the results are averaged over the range o f energies b e t w e e n - 0 . 7 eV and - 0 . 1 eV below the Fermi energy (fig. 4), the derived w o r k f u n c t i o n is 4.25 V which agrees with their results o f 4.16 V and 4.34 V m o s t likely because their data were averaged over a similar energy range. The data for the (111) face reveal a systematic d e p e n d e n c e o f q51 on the value o f ~ In R~ 3e that is similar to the (112) result. In this case, however, most o f the points (4~1, ~ lnR/Oe) lie on the left side o f the origin. For this reason and because
T.V. Forburger et al./Field emission work functions
429
Table 3 Summary of tungsten work functions in volts determined by various methods (column 3 contains results for the field emission retarding potential method; column 4 contains average values from several thermionic emission experiments discussed by Rlvlere " '" in his 1969 review; column 5 gives the contact potential difference results of Hopkins and Pender also discussed by Rivi~re) W crystal face
Present work
FERP (Ref. 13)
Thermionic emission (Ref. 14)
CPD (Ref. 14)
(100)
4.57 ± 0.14
4.63 ± 0.02
4.58 ± 0.08
4.67 + 0.04
(112)
4.89 ± 0.15
(013)
5.19 -+ 0.16
4.80 ± 0.05 a 5.24 -+ 0.05
(110) (111)
4.83 -+ 0.38
5.25 ± 0.02
5.30 -+ 0.12
4.47 ± 0.02
4.40 -+ 0.03
5.05 -+ 0.02 5.15 ± 0.03
a Rivi~re did not calculate an average value for the (112) face since the two results shown are in disagreement.
of the fact that the (111) data were taken for only four different electric fields the second extrapolation yields a value for 92 which has a large statistical uncertainty.
j
I
i
I
r
I
I
6.0
5.0
"~ 4.0
s
3.C
I
-0.8
I
-0.7
I
-0.6
I
-0.5 ==E-El
I
-0,4
I
-0.3
I
-0.2
-0.1
(eV)
Fig. 4. Single field determinations Oe for W (013) versus energy. The filled circles are values for q5e measured over a 0.1 eV energy interval centered about the energy indicated. Each open circle corresponds to an average value for Oe taken over the entire range between - 0 . 1 eV and the energy indicated. The squares show the values of ¢~e measured by Swanson and Crouser for the (013) plane.
430
T.V. Vorburger et al./FieM emission work Junctions
The final result of 4.83 + 0.38 V is higher than that obtained by other methods, but this difference is not significant in view o f the large uncertainty.
4. Conclusions Previous determinations of the absolute value of the work function from field emission total current and energy distribution measurements have produced contradictory results. The determinations were based on the Fowler-Nordheim equation which relates the field emission current of a free electron metal to its work function. We have shown that "band structure" effects have a sufficiently important effect on the field emission energy distribution that they must be taken into account if correct (and consistent) values o f tile work function are to be obtained. We have taken "band structure" effects, caused by tire periodic lattice potential, into account by writing j'(e) = R(e)/'~(e) where/'(e),J'o(e ) are the field emission currents at energy e of the actual metal and o f the free electron-like metal respectively. A theoretical analysis has produced an expression for R(e). This expression shows that R depends not only on energy but for some faces has a dependence on electric field as well. The dependence of R on both energy and field must be considered in order to determine the work function, and a procedure for doing this has been given in the text. This technique was shown to give reliable absolute work functions for the (100) and (112) planes o f a W emitter. The value obtained for (013) W was much higher than what we would expect and preliminary data indicates a similar deviation exists on the (111) face. It is possible that there is an error in the high index directions, that is caused by the roughness of the surface, which is not adequately represented by our one dimensional model. Specifically, tile equation for determining 0 [eq. (11)] is based on the expression for tile free electron tunneling probability, eq. (8b), which is derived for a one-dimensional tunneling model. If this model were not valid, because of surface roughness on an atomic scale or because of the surface curvature, eq. (8b) would be incorrect. In such a case, one would expect to observe such phenomena as the shifting of the (111) structure with increasing electric fields. Alternatively, since the (111) and (013) are not well developed planes the probe hole could accept sufficient electron emission from surrounding regions (which may have different work functions and may be experiencing different electric fields) to cause some error in the results [22]. It is not felt that this edge emission is affecting the results for the (100) and (112) faces because of the large sizes of these planes and because the results agree with those obtained by other methods. It might be possible to determine whether any or all of these possible effects is significant in the high index directions by taking work function measurements for emitters with varying sizes or by varying the sizes of the planes themselves using field evaporation [22]. This technique for measuring work functions is relatively complicated, but it
T. II. Vorburger et al./FieM emission work ]unctions
431
does have the capability of measuring absolute (not relative) work functions for the low index planes which are well developed on a field emitter [(110) [15], (100), (112) for W]. We view this technique as primarily useful for field emission energy distribution studies of materials where reliable absolute work functions are not available. For example, to apply the formulation of Penn and Plummer [ 16] to measure the "surface density of states" a free electron energy distribution must be calculated first for the appropriate work function and field. The work function can be obtained for well developed planes by applying the above techniques, and the field can then be calculated from the slope of the Fowler-Nordheim plot [15].
Acknowledgements The authors are grateful to Drs. C.J. Powell, C.E. Kuyatt, and A.J. Melmed for a number of helpful discussions.
References [1] [21 [3] [4] [5]
E.W. M~ller, J. Appl. Phys. 26 (1955) 732. R.H. Fowler and L. Nordheim, Proc. Roy. Soc. (London) A 119 (1928) 173. H.C. Miller, J. Franklin Inst. 282 (1966) 382. T. Engel and R. Gomer, J. Chem. Phys. 52 (1970) 5572. L.W. Swanson and A.E. Bell, Advan. Electron. Electron Phys. 32 (1973) 193. [6] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487. [7] C.J. Todd and T.N. Rhodin, Surface Sci. 36 (1973) 353. [8] R.D. Young and E.W. M~ller, J. Appl. Phys. 33 (1962) 91. [9] R.D. Young and H.E. Clark, Phys. Rev. Letters 17 (1966) 351; R.D. Young and H.E. Clark, Appl. Phys. Letters 9 (1966) 265. [I0] R.D. Young, Phys. Rev. 113 (1959) 110. [11] L.W. Swanson and L.C. Crouser, Phys. Rev. 163 (1967) 622. [12] C. Lea and R. Gomer, J. Chem. Phys. 54 (1971) 3349. [13] R.W. Strayer, W. Mackie and L.W. Swanson, Surface Sci. 34 (1973) 225. [14] J.C. Rivi~re, in: Solid State Surface Science, Vol. 1, Ed. M. Green (Dekker, New York, 1969) p. 179. [15] E.W. Plummer and A.E. Bell, J. Vacuum Sci. Technol. 9 (1972) 583. [16] D. Penn and E.W. Plummer, Phys. Rev. B9 (1974) 1216. [17] C.E. Kuyatt and E.W. Plummer, Rev. Sci. Instr. 43 (1972) 108. [18 ] Estimates of the emitter radius were based on considerations given by R.H. Good and E.W. M~ller, in: Handbuch der Physik, Vol. 21, Ed. S. Fl~gge (Springer, Berlin, 1956) p. 176. [191 J.A. Becker, Bell Syst. Tech. J. 30 (1951) 907. [20] M. Drechsler and H. Liepack, in: Adsorption et Croissance Cristalline (CNRS, Paris, 1965) p. 49. [21 ] C.E. Mendenhall and C.F. de Voe, Phys. Rev. 51 (1937) 346. [22] A.J. Melmed, 20th Field Emission Symposium, Pennsylvania State University, University Park, Pennsylvania, August 1973 (unpublished). This author has measured the slope of Fowler-Nordheim plots for W (110) and Ru (1010) as a function of plane size and observed variations in the derived work function of several percent.