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Fowler–Nordheim theory for a spherical emitting surface
Ultramicroscopy 95 (2003) 49–56
Fowler–Nordheim theory for a spherical emitting surface C.J. Edgcombea,b,* a
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK b Granta Electronics Ltd., 25 St. Peter’s Road, Coton, CB3 7PR, UK Received 1 August 2001; received in revised form 6 January 2002
Abstract In tests on a field emitter whose dimensions and work function were known, Fowler–Nordheim (F–N) theory as usually stated for a planar emitter was found to give poor agreement with observations. The effect of curvature of the emitting surface has been modelled by including (a) non-linear variation of potential with distance from the surface; (b) the consequent changes in the exponent and pre-exponential terms in the F–N expression for current; and (c) the variation of current density over the surface, modelled by an effective solid angle. Application of the resulting expression to the measured data gives estimates for apex radius which agree much more closely with the measured value than the value from planar theory does. r 2002 Elsevier Science B.V. All rights reserved. PACS: 79.70.+q; 81.05.Uw; 81.15.Jj; 85.45.Bz; 85.45.Db Keywords: Electron sources; Tunneling theory
1. Introduction The process of deposition of carbon by e-beam on rigid substrates has been developed by Antognozzi et al. [1] to produce tapered tips usable as field emitters. The dimensions of an individual tip can be controlled, and can be measured together with its current–voltage characteristics. Such emitters have been measured in situ [2] as having typically length (L) of 1 mm and apex radius (a) of 5 nm, and have been deposited on smooth tungsten balls of radius about 65 mm. The work *Corresponding author. Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK. Tel.: +44-01223-337333; fax: +44-01223-363263. E-mail address: [email protected] (C.J. Edgcombe).
function f of the tip material was measured independently by use of a Kelvin probe [3]. The field factor b; equal to the ratio of the field F at the apex surface to the anode–cathode voltage Va ; was found by finite-element computation [2]. Measured values of current and voltage for a single tip lie on a straight line on the conventional Fowler– Nordheim (F–N) plot (Fig. 1), but quantities calculated from the slope and intercept do not agree well with the experimental values. Analysis using conventional F–N theory showed that if the field at the emitter was that suggested by calculation, the work function would be only about 66% of the value found by independent measurement. Alternatively, if the work function for field emission is taken as the value found by Kelvin probe microscopy, then the actual field at
0304-3991/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 2 ) 0 0 2 9 6 - 6
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
50 -30.5
-2
-32.0
ln (I/V /AV )
-31.5
2
-31.0
-32.5 y = -2.5717x - 16.024 R2 = 0.9983
-33.0 -33.5 5.6
5.8
6.0
6.2
6.4
6.6
6.8
1/V /10-4V-1
Fig. 1. Conventional Fowler–Nordheim plot for carbon tip.
the apex appears to be about 1.9 times the calculated value. Thus for this sample, the value of (f3=2 =F ) found from the F–N slope using planar theory was 0.54 of the value found by other measurements. Also, for a wide range of values of work function and with the apex field set for consistency with the observed slope of the F–N plot, estimates of the apex radius gave a value about 2.6 times the observed radius. There are clearly many uncertainties and inaccuracies in attempting to apply planar F–N theory to an emitting surface which is typically a spherical cap. The following sections show how the curvature of the emitting surface modifies the theory, and provides more accurate results for the tested sample.
2. Potential near spherical surface We computed the potential around a tip of length L ¼ 2 mm, apex radius a ¼ 10 nm and shank angle 101, supported on a base of radius 65 mm, near a hollow anode. We find that near the axis and at radius r from the centre of curvature, with r slightly greater than a; the electric field is proportional to r2 ; even though the cap is supported on a long shank. The analytic Laplace solution for potential near the apex of a hemi-ellipsoid on a plane agrees with this observation.
For calculation of the transmission factor, the ‘image’ potential due to exchange and correlation effects should also be included in the total potential. It should ideally be correct for a geometry like that of the tip, such as a hemiellipsoid on a plane base. For first trial we have used the image potential for a spherical conductor which coincides with the emitter surface near the axis. Hence the potential due to work function f, applied field and image is approximated here by Ve1 ¼ Ef þ f eFað1 a=rÞ e2 a=8pe0 ðr2 a2 Þ ðr > aÞ; where Ef is the Fermi energy, e is the magnitude of electronic charge, F is the field at the emitter surface (in absence of image effect), and a is the radius of the emitter. The outer radius b of the barrier is defined as the radius at which the electron’s potential energy equals its total energy (both measured from the same reference level). For an electron at the Fermi level, in the absence of image potential, the barrier thickness (b a) is given by b a ¼ ax=ð1 xÞ ¼ fa=ðaeF fÞ: It is greater than that for a planar barrier with the same F by the factor 1=ð1 f= aeF Þ ¼ 1=ð1 xÞ (see Fig. 2), and it changes with F slightly faster than for planar geometry. These small differences have a large effect on the variation of current with field, since they multiply the exponent in the transmission factor.
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
51
4. Pre-exponential term and current density
Energy
φ
Spherical
Ef a
Planar
b
r
Fig. 2. Potential in absence of image effect.
3. Transmission factor
The emitted current density is the integral of the product of the transmission factor and a supply function over Un ; the energy of incidence at the surface. Hence the expression appearing before the exponential in the complete expression for the current density is related to the exponent, as shown by Hawkes and Kasper [5]. When the exponent is changed by modification of the surface shape, the pre-exponential expression changes also. The following summary corresponds to the derivation of Hawkes and Kasper for planar theory. The transmission factor is approximated around En ¼ 0 by
A derivation of the transmission factor DðUn Þ for an electron incident on a barrier with energy Un corresponding to its normal momentum is given in Appendix A. The result is
DðUn ÞED0 expðEn =dÞ;
DðUn Þ ¼ exp½2DGEexp½f2ð2mÞ1=2 =_g Z
Re ½V ðrÞ Un 1=2 dr;
where the parameters D0 and d are used like those given by Hawkes and Kasper (Section 44.4), but are defined here by D0 ¼ DðEn ¼ 0Þ ¼ exp½c2 f3=2 f ðx; qÞ=Fx;
where for the spherical system, d 1 ¼ ðqDG=qUn ÞEn ¼0 ¼ 3c2 f1=2 wðx; qÞ=2Fx;
V ðzÞ Un ¼ f eF0 að1 a=rÞ e2 a=8pe0 ðr2 a2 Þ Un þ Ef : The integration is less straightforward than for the planar case. We write the result as 3=2
DðUn Þ ¼ exp½c2 ðf En Þ
¼ exp½c2 aeðf En Þ
f ðxE ; qÞ=FxE
1=2
f ðxE ; qÞ;
where En ¼ Un Ef ; xE ¼ ðf En Þ=aeF ; q ¼ e2 = 8pe0 af; c2 ¼ 4ð2 mÞ1=2 =ð3e_Þ and Z f ðxE ; qÞ ¼ ð3=2Þ ½1 ð1=xE Þð1 1=rÞ q=ðr2 1Þ1=2 dr: The function f ðx; qÞ is greater than (x 2q) for all realistic values of x and q tested thus far. Its equivalent in the planar theory is the product ½xvðyÞ; but there vðyÞ is typically B0.6 and x-0 since the radius of the surface is infinite. Also, qf =qx is typically 1.1–1.6 which is greater than the planar equivalent, qðvxÞ=qxE0:93: An approximation for f is given in Appendix B.
where x ¼ f=aeF and wðx; qÞ ¼ ðf þ 2x@f =@x 2q@f =@qÞ=3x: The function w defined here differs from the function g used previously [3]; the advantage of using w is that it shows more directly the effect on the pre-exponential of changing from planar to spherical theory. Approximations for w are given in Appendix B. In previous expositions of the theory, such as those by Good and Muller . [4] and by Hawkes and Kasper, the density of states gðEÞ is taken as constant over the range of integration. The electrons contributing most to the current are those emitted within a few kT of the Fermi level. In the free-electron model, gðEÞ is proportional to ðE=EF Þ1=2 : So the relative variation of gðEÞ near the Fermi level due to a change in E of 7kT is7(1/2) kT=Ef ; which is 7 0.5% for T ¼ 300 K and a Fermi energy of 5 eV. The approximation of taking gðEÞ as constant seems likely to be adequate for many purposes.
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
52
With gðEÞ treated as constant, the result of integrating the transmitted current over Un is that the current density (in the absence of temperature enhancement) can be expressed as
without making implicit assumptions about the radius. From (2), O is given by Z ð3Þ O ¼ JðyÞ2p sin y dy=J0 :
J ¼ ð4pem=h3 Þd2 exp½D0 ¼ ½c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 f3=2 f ðx; qÞ=Fx ð1aÞ ¼ ½c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 aef1=2 f ðx; qÞ; ð1bÞ
An approximate evaluation of (3) leads to OE 8p½ð5 S=Va Þ1 2x0 ð4 S=Va Þ1 þ x20 ð3 S=Va Þ1 ð1 x0 Þ2 ;
3
where c1 ¼ ðe =8phÞ: In spherical geometry f and w vary more rapidly than their planar equivalents; also the magnitude of the exponent is greater, and the expression before the exponential smaller, than for planar geometry. The function wðx; qÞ replaces the function tðyÞ used in the planar theory. In many cases the anode–cathode voltage Va is known more accurately than the surface field. The parameter S is defined here as the derivative of the exponent in (1) with respect to Va1 : S ¼ c2 f3=2 f 0 ðx; qÞ=b; where f 0 ðx; qÞ is qf ðx; qÞ=qx and F has been replaced by bVa : S thus represents the rate of change with Va1 of only the exponent in (1a) or (1b).
ð4Þ
where S is evaluated at the anode voltage Va and corresponding x: The effective solid angle given by (4) is plotted in Fig. 3 as a function of F0 for the parameters b ¼ 3:27 106 V m1 and Sm ¼ 23364 V. At field strengths typical of field emission, the solid angle increases approximately as F00:73 : The total current can now be written from (1) and (2) as I ¼ ½Oa2 c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 f3=2 f ðx; qÞ=ðFxÞ ¼ ½Oa2 c1 F 2 =ðfw2 ðx; qÞÞexp½c2 f1=2 aef ðx; qÞ: ð5Þ Use of the approximation w ðx; qÞE1=ð1 xÞ4=3 leads to
5. Effective solid angle and total current
I ¼ Oa2 J0 :
ð2Þ
The benefit of defining an angle rather than an emitting area is that the angle includes the effect of variation of J over the surface, but depends much less on the radius of the emitter. This is convenient in extraction of data from the F–N plot, allowing the variation of J over the surface to be modelled
which shows that the expression before the exponential contains the parameters a; F and f as for planar geometry, but also includes the solid angle O and a factor like ð1 xÞ8=3 : 10.0
Steradians
In practice, the field near the apex of a real tip varies over the surface, so the current density also varies greatly over the emitting surface. It is helpful to express the total current as a multiple of the maximum current density on the cap. This can be done by defining an effective solid angle, O; as the ratio of the total current I to the product of the square of the radius a of the emitting surface and the maximum current density J0 (on the axis, due to the corresponding maximum field F0 ):
IE½Oa2 c1 F 2 ð1 xÞ8=3 =f exp½c2 f1=2 aef ðx; qÞ
Using F = F 0 cos θ /2 Experimental range
1.0 y=1 Emission threshold
0.1 1
10 F0 /GVm-1
Fig. 3. Variation of effective solid angle with field.
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C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
6. Extraction of parameters from F–N plot To use (5) in the same way as for the planar theory, we might consider plotting Ln(w2 I=OVa2 ) against f ðx; qÞ; but this is not feasible when we do not know the ratio x=Va or q for the experimental conditions. Some information can be obtained from the graph of LnðIÞ against 1=Va : Assuming that a smooth curve can be drawn through the experimental points, the tangent to the curve can be drawn at any point, and its slope S1 and intercept Y1 can be determined. From (5), theoretical values of these are S1 ¼ dLnðIÞ=dVa1 ¼ ðO0 x=O 2w0 x=w 2ÞVa þ S;
ð6Þ
Y1 ¼ LnðIÞ S1 =Va ¼ LnðPÞ þ T=Va S1 =Va ;
ð7Þ
where P ¼ Oc1 f=ðe2 w2 x2 Þ;
53
Independent measurement on the material of one tip characterised in detail showed that the work function was close to 5.0 eV. We have taken both this value and the observed current/voltage relation as given, and have investigated how well the theory above fits these observations. The plot of LnðIÞ against Va1 has a slope S1 of 28953 V and an intercept Y1 of 0.757, at a mean anode voltage of 1625 V. In the case tested, with x in the range 0.1–0.3, iterative solution of (6) provided a consistent set of S; b; a and q for each x: These values enabled P to be calculated by both methods. Values of b; a and F are plotted in Fig. 4(a), to show how these quantities vary with x: For this tip, S1 is about 1.3 times S: The ratio Pexp =Pfe is plotted against x in Fig. 4(b). It can be seen that the experimental value of P approaches the free-electron value most closely where x is about 0.15. This value of x provides best consistency between theory and experiment, if it is assumed that the density of
ð8Þ
T ¼ c2 f3=2 f =bx: The specific value of x for the experimental conditions is not known directly. However, all the quantities except S1 ; Va and Y1 in (6) and (7) can be calculated for given ranges of values of x and q: Some iteration is desirable at this stage to obtain values of a and q consistent with each x: Then for each set of parameters, P can be calculated in two ways. One is from (7), by use of the experimental values of S1 ; Va and Y1 and the calculated T: We denote this value of P by Pexp : Another value of P may be obtained by calculation from (8), and this is denoted by Pfe : We can compare the two calculated values by plotting Pexp =Pfe as a function of x: The calculation of Pfe from (8) assumes that the emitter has the density of states of free-electron theory. If the density of states near the Fermi level is appreciably less, as might occur for a semiconductor emitter, then the value of Pfe given by (8) will be an overestimate. In this case the ratio Pexp =Pfe cannot be expected to approach unity closely.
Fig. 4. (a) Values of b; a and F ; and (b) ratio Pexp =Pfe ; both obtained for tested tip.
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C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
states near the Fermi energy for the tested carbon tip is similar to that for the free-electron model. A further check on consistency is available for the present experiment by comparing the observed value of emitter radius with that predicted at the value of x for maximum Pexp =Pfe : From this value of x; the emitting radius and voltage gradient are calculated as 5.2 nm and 6.5 109 V m1(at the mid-range anode voltage). This value of emitter radius is in good agreement with the observed apex radius of 5 nm. Thus, in this case, the value of x giving agreement between observed and calculated emitter radii also gives closest agreement with the free-electron model. This suggests that the density of states at the Fermi level for the tip tested is within a few per cent of the free-electron value. Although the density of states for the emitter is not known accurately, it is clear that in this case the spherical theory enables consistent sets of parameters to be deduced which correspond to realistic values of emitting radius and surface field. The surface field F deduced in this way from the measured F–N slope and work function is greater than that calculated from the measured tip dimensions by a factor of about 2.6. This could be due to roughness of the emitting surface on a scale smaller than the apex radius, that is on a near-atomic scale. This seems possible in view of the method of deposition of the tip. Other possibilities which do not account for the discrepancies in work function or apex radius include: (a) error in f (for deduction of apex radius); (b) use of planar rather than spherical coordinate system and (c) neglect of variation of emission over the surface of the emitter.
The effective solid angle of emission varies approximately as (applied voltage)0.73 in the field-emitting range. The field strength at a spherical emitting surface is larger than would be estimated by use of planar theory with the conventional F–N plot. Revised expressions have been found for the slope and intercept of a tangent to the graph of Ln(I)Va1 ; but these do not give the work function or field directly. Consistent sets of parameters can be found as functions of x: For the single carbon tip tested, with separately measured work function: Use of planar theory with field calculated from geometry gives a tip radius about 2.6 times that observed, for a wide range of work function. The best fit of spherical theory with experimental data gives a value of tip radius in agreement with the measured value, and surface field about 2.6 times the calculated value. The method of fabrication could have produced atomic-scale surface roughness, leading to a field enhancement of the order of 3 relative to a smooth surface. For this experiment, spherical theory fits the observations much better than planar theory.
7. Conclusions
The transmission factor D for an electron incident on the barrier with energy E is determined by the potential distribution in the barrier region. A substitution for the spatially varying part of the wave function
The rate of change of barrier thickness with applied field is slightly greater for a spherical emitter than for a planar one, for a given field at the surface. For a spherical emitting cap, the image correction factor vðyÞ used in planar theory should be replaced by ½f ðx; qÞ=x in the exponent of the transmission factor, with f ; q and x as defined above.
Acknowledgements The author thanks Prof. L. M. Brown and Prof. U. Valdre" for many stimulating discussions
Appendix A. Transmission factor for potential barrier
c ¼ expðGÞ;
ðA:1Þ
(where G may be complex) allows the time. independent Schrodinger equation to be written as rG rG r2 G ¼ ð2m=_2 ÞðV EÞ:
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
55
(proportional to jc22 =c21 j) is approximated by
So, if jr2 Gj5ð2m=_2 ÞjV Ej
ðA:2Þ
then
D ¼ jexpð2 DGÞj Z Eexp ð2ð2mÞ1=2 =_ÞRe ðV Un Þ1=2 dr :
rG rGEð2m=_2 ÞðV EÞ:
ðA:3Þ
The vector =G can be written as the sum of vectors normal and tangential to the local equipotential surface: =G ¼ ð=GÞn þ ð=GÞt
Also, since the direction of =G is nearly radial, r2 GEðð2mÞ1=2 =_r2 Þqðr2 ðV EÞ1=2 Þ=qr: Hence for condition (A.2) to be met at any point in the region where ðV EÞ > 0; it is necessary that
from which
jðq=qrÞðr2 ðV EÞ1=2 Þj5ðð2mÞ1=2 =_Þr2 ðV EÞ:
rG rG ¼ ðrGÞ2n þ ðrGÞ2t : We now define Un and Ut ; two values of energy for a single electron, which sum to the total energy E of the electron and are related to the corresponding =G for that electron and local potential energy V by Un ¼ V ð_2 =2mÞðrGÞ2n ; Ut ¼ ð_2 =2mÞðrGÞ2t : The definition (A.1) shows that the ratio of values of c at positions 1 and 2 is c2 =c1 ¼ eG2 =eG1 ¼ eDG ; where
ðA:4Þ With the experimental tip considered, condition (A.4) is met for ðV 2EÞB5 eV, but not for (V EÞB1 eV, when the two sides of (A.4) become comparable in magnitude. This produces some inaccuracy when D is evaluated by use of (A.3). For greater accuracy where (V E) is small, it would be preferable to improve the approximation for rG or use Airy functions. However, to demonstrate simply the behaviour of the complete barrier, the expression (A.3) will be used here as an approximation for D:
DG ¼ G2 G1 : The change DG over a defined path is given by Z DG ¼ rG ds: So on choosing a path normal to the equipotentials, with coordinate n; through a region of varying V ; DG is given by DG ¼
Z
1=2
ðrGÞn dn ¼ ðð2mÞ
Z =_Þ
ðV Un Þ1=2 dn:
To find the change in amplitude of c caused by the barrier, we take the limits of integration as the bounding values of n at which V 2Un is zero. For the geometry of the emitting tip, the equipotential of the cathode surface is taken to be spherical, while those for greater voltage have slightly different shapes. Within the barrier thickness, the variation of the equipotentials from spherical is small. We therefore approximate by taking the direction of the path as radial, when the transmission function
Appendix B. Approximations for the functions f and w The following approximations have been obtained by fitting simple expressions to calculated values of f ðx; 0Þ and wðx; 0Þ; and by estimating the correction needed to account for non-zero q: The values of q found in practice have been less than 0.1. f ðx; 0Þ; exact solution: (3/2)[arcsin x1=2 =ðx x2 Þ1=2 1 approximation: 2½ð1 xÞ3=2 1 f ðx; qÞ; approximation: f ðx; 0Þ 2q (this implies that @f ðx; qÞ=@x is independent of q which is not exactly correct). w(x,0) is given exactly by ½f ðx; 0Þ þ 2xqf ðx; 0Þ= qx=3x approximation: 1=ð1 xÞ4=3 w ðx; qÞ is given exactly by ½f ðx; qÞ þ 2xqf ðx; qÞ= qx 2qqf ðx; qÞ=qq=3x approximation: wðx; 0Þ þ 4q=3x
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C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
References [1] M. Antognozzi, A. Sentimenti, U. Valdr"e, Microsc. Microanal. Microstruct. 8 (1997) 355. [2] C. Edgcombe, U. Valdr"e, J. Microsc. 203 (2001) 188.
[3] C. Edgcombe, U. Valdr"e, Phil. Mag. B 82 (2002) 1009. [4] R.H. Good, E.W. Muller, . in: S. Flugge . (Ed.), Handbuch der Physik, Berlin: Springer, 1956, pp. 156–231. [5] P. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 2, Academic Press, New York, 1989 (Chapter 44).
Fowler–Nordheim theory for a spherical emitting surface C.J. Edgcombea,b,* a
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK b Granta Electronics Ltd., 25 St. Peter’s Road, Coton, CB3 7PR, UK Received 1 August 2001; received in revised form 6 January 2002
Abstract In tests on a field emitter whose dimensions and work function were known, Fowler–Nordheim (F–N) theory as usually stated for a planar emitter was found to give poor agreement with observations. The effect of curvature of the emitting surface has been modelled by including (a) non-linear variation of potential with distance from the surface; (b) the consequent changes in the exponent and pre-exponential terms in the F–N expression for current; and (c) the variation of current density over the surface, modelled by an effective solid angle. Application of the resulting expression to the measured data gives estimates for apex radius which agree much more closely with the measured value than the value from planar theory does. r 2002 Elsevier Science B.V. All rights reserved. PACS: 79.70.+q; 81.05.Uw; 81.15.Jj; 85.45.Bz; 85.45.Db Keywords: Electron sources; Tunneling theory
1. Introduction The process of deposition of carbon by e-beam on rigid substrates has been developed by Antognozzi et al. [1] to produce tapered tips usable as field emitters. The dimensions of an individual tip can be controlled, and can be measured together with its current–voltage characteristics. Such emitters have been measured in situ [2] as having typically length (L) of 1 mm and apex radius (a) of 5 nm, and have been deposited on smooth tungsten balls of radius about 65 mm. The work *Corresponding author. Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, UK. Tel.: +44-01223-337333; fax: +44-01223-363263. E-mail address: [email protected] (C.J. Edgcombe).
function f of the tip material was measured independently by use of a Kelvin probe [3]. The field factor b; equal to the ratio of the field F at the apex surface to the anode–cathode voltage Va ; was found by finite-element computation [2]. Measured values of current and voltage for a single tip lie on a straight line on the conventional Fowler– Nordheim (F–N) plot (Fig. 1), but quantities calculated from the slope and intercept do not agree well with the experimental values. Analysis using conventional F–N theory showed that if the field at the emitter was that suggested by calculation, the work function would be only about 66% of the value found by independent measurement. Alternatively, if the work function for field emission is taken as the value found by Kelvin probe microscopy, then the actual field at
0304-3991/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 2 ) 0 0 2 9 6 - 6
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
50 -30.5
-2
-32.0
ln (I/V /AV )
-31.5
2
-31.0
-32.5 y = -2.5717x - 16.024 R2 = 0.9983
-33.0 -33.5 5.6
5.8
6.0
6.2
6.4
6.6
6.8
1/V /10-4V-1
Fig. 1. Conventional Fowler–Nordheim plot for carbon tip.
the apex appears to be about 1.9 times the calculated value. Thus for this sample, the value of (f3=2 =F ) found from the F–N slope using planar theory was 0.54 of the value found by other measurements. Also, for a wide range of values of work function and with the apex field set for consistency with the observed slope of the F–N plot, estimates of the apex radius gave a value about 2.6 times the observed radius. There are clearly many uncertainties and inaccuracies in attempting to apply planar F–N theory to an emitting surface which is typically a spherical cap. The following sections show how the curvature of the emitting surface modifies the theory, and provides more accurate results for the tested sample.
2. Potential near spherical surface We computed the potential around a tip of length L ¼ 2 mm, apex radius a ¼ 10 nm and shank angle 101, supported on a base of radius 65 mm, near a hollow anode. We find that near the axis and at radius r from the centre of curvature, with r slightly greater than a; the electric field is proportional to r2 ; even though the cap is supported on a long shank. The analytic Laplace solution for potential near the apex of a hemi-ellipsoid on a plane agrees with this observation.
For calculation of the transmission factor, the ‘image’ potential due to exchange and correlation effects should also be included in the total potential. It should ideally be correct for a geometry like that of the tip, such as a hemiellipsoid on a plane base. For first trial we have used the image potential for a spherical conductor which coincides with the emitter surface near the axis. Hence the potential due to work function f, applied field and image is approximated here by Ve1 ¼ Ef þ f eFað1 a=rÞ e2 a=8pe0 ðr2 a2 Þ ðr > aÞ; where Ef is the Fermi energy, e is the magnitude of electronic charge, F is the field at the emitter surface (in absence of image effect), and a is the radius of the emitter. The outer radius b of the barrier is defined as the radius at which the electron’s potential energy equals its total energy (both measured from the same reference level). For an electron at the Fermi level, in the absence of image potential, the barrier thickness (b a) is given by b a ¼ ax=ð1 xÞ ¼ fa=ðaeF fÞ: It is greater than that for a planar barrier with the same F by the factor 1=ð1 f= aeF Þ ¼ 1=ð1 xÞ (see Fig. 2), and it changes with F slightly faster than for planar geometry. These small differences have a large effect on the variation of current with field, since they multiply the exponent in the transmission factor.
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
51
4. Pre-exponential term and current density
Energy
φ
Spherical
Ef a
Planar
b
r
Fig. 2. Potential in absence of image effect.
3. Transmission factor
The emitted current density is the integral of the product of the transmission factor and a supply function over Un ; the energy of incidence at the surface. Hence the expression appearing before the exponential in the complete expression for the current density is related to the exponent, as shown by Hawkes and Kasper [5]. When the exponent is changed by modification of the surface shape, the pre-exponential expression changes also. The following summary corresponds to the derivation of Hawkes and Kasper for planar theory. The transmission factor is approximated around En ¼ 0 by
A derivation of the transmission factor DðUn Þ for an electron incident on a barrier with energy Un corresponding to its normal momentum is given in Appendix A. The result is
DðUn ÞED0 expðEn =dÞ;
DðUn Þ ¼ exp½2DGEexp½f2ð2mÞ1=2 =_g Z
Re ½V ðrÞ Un 1=2 dr;
where the parameters D0 and d are used like those given by Hawkes and Kasper (Section 44.4), but are defined here by D0 ¼ DðEn ¼ 0Þ ¼ exp½c2 f3=2 f ðx; qÞ=Fx;
where for the spherical system, d 1 ¼ ðqDG=qUn ÞEn ¼0 ¼ 3c2 f1=2 wðx; qÞ=2Fx;
V ðzÞ Un ¼ f eF0 að1 a=rÞ e2 a=8pe0 ðr2 a2 Þ Un þ Ef : The integration is less straightforward than for the planar case. We write the result as 3=2
DðUn Þ ¼ exp½c2 ðf En Þ
¼ exp½c2 aeðf En Þ
f ðxE ; qÞ=FxE
1=2
f ðxE ; qÞ;
where En ¼ Un Ef ; xE ¼ ðf En Þ=aeF ; q ¼ e2 = 8pe0 af; c2 ¼ 4ð2 mÞ1=2 =ð3e_Þ and Z f ðxE ; qÞ ¼ ð3=2Þ ½1 ð1=xE Þð1 1=rÞ q=ðr2 1Þ1=2 dr: The function f ðx; qÞ is greater than (x 2q) for all realistic values of x and q tested thus far. Its equivalent in the planar theory is the product ½xvðyÞ; but there vðyÞ is typically B0.6 and x-0 since the radius of the surface is infinite. Also, qf =qx is typically 1.1–1.6 which is greater than the planar equivalent, qðvxÞ=qxE0:93: An approximation for f is given in Appendix B.
where x ¼ f=aeF and wðx; qÞ ¼ ðf þ 2x@f =@x 2q@f =@qÞ=3x: The function w defined here differs from the function g used previously [3]; the advantage of using w is that it shows more directly the effect on the pre-exponential of changing from planar to spherical theory. Approximations for w are given in Appendix B. In previous expositions of the theory, such as those by Good and Muller . [4] and by Hawkes and Kasper, the density of states gðEÞ is taken as constant over the range of integration. The electrons contributing most to the current are those emitted within a few kT of the Fermi level. In the free-electron model, gðEÞ is proportional to ðE=EF Þ1=2 : So the relative variation of gðEÞ near the Fermi level due to a change in E of 7kT is7(1/2) kT=Ef ; which is 7 0.5% for T ¼ 300 K and a Fermi energy of 5 eV. The approximation of taking gðEÞ as constant seems likely to be adequate for many purposes.
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With gðEÞ treated as constant, the result of integrating the transmitted current over Un is that the current density (in the absence of temperature enhancement) can be expressed as
without making implicit assumptions about the radius. From (2), O is given by Z ð3Þ O ¼ JðyÞ2p sin y dy=J0 :
J ¼ ð4pem=h3 Þd2 exp½D0 ¼ ½c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 f3=2 f ðx; qÞ=Fx ð1aÞ ¼ ½c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 aef1=2 f ðx; qÞ; ð1bÞ
An approximate evaluation of (3) leads to OE 8p½ð5 S=Va Þ1 2x0 ð4 S=Va Þ1 þ x20 ð3 S=Va Þ1 ð1 x0 Þ2 ;
3
where c1 ¼ ðe =8phÞ: In spherical geometry f and w vary more rapidly than their planar equivalents; also the magnitude of the exponent is greater, and the expression before the exponential smaller, than for planar geometry. The function wðx; qÞ replaces the function tðyÞ used in the planar theory. In many cases the anode–cathode voltage Va is known more accurately than the surface field. The parameter S is defined here as the derivative of the exponent in (1) with respect to Va1 : S ¼ c2 f3=2 f 0 ðx; qÞ=b; where f 0 ðx; qÞ is qf ðx; qÞ=qx and F has been replaced by bVa : S thus represents the rate of change with Va1 of only the exponent in (1a) or (1b).
ð4Þ
where S is evaluated at the anode voltage Va and corresponding x: The effective solid angle given by (4) is plotted in Fig. 3 as a function of F0 for the parameters b ¼ 3:27 106 V m1 and Sm ¼ 23364 V. At field strengths typical of field emission, the solid angle increases approximately as F00:73 : The total current can now be written from (1) and (2) as I ¼ ½Oa2 c1 F 2 =ðfw2 ðx; qÞÞ exp½c2 f3=2 f ðx; qÞ=ðFxÞ ¼ ½Oa2 c1 F 2 =ðfw2 ðx; qÞÞexp½c2 f1=2 aef ðx; qÞ: ð5Þ Use of the approximation w ðx; qÞE1=ð1 xÞ4=3 leads to
5. Effective solid angle and total current
I ¼ Oa2 J0 :
ð2Þ
The benefit of defining an angle rather than an emitting area is that the angle includes the effect of variation of J over the surface, but depends much less on the radius of the emitter. This is convenient in extraction of data from the F–N plot, allowing the variation of J over the surface to be modelled
which shows that the expression before the exponential contains the parameters a; F and f as for planar geometry, but also includes the solid angle O and a factor like ð1 xÞ8=3 : 10.0
Steradians
In practice, the field near the apex of a real tip varies over the surface, so the current density also varies greatly over the emitting surface. It is helpful to express the total current as a multiple of the maximum current density on the cap. This can be done by defining an effective solid angle, O; as the ratio of the total current I to the product of the square of the radius a of the emitting surface and the maximum current density J0 (on the axis, due to the corresponding maximum field F0 ):
IE½Oa2 c1 F 2 ð1 xÞ8=3 =f exp½c2 f1=2 aef ðx; qÞ
Using F = F 0 cos θ /2 Experimental range
1.0 y=1 Emission threshold
0.1 1
10 F0 /GVm-1
Fig. 3. Variation of effective solid angle with field.
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6. Extraction of parameters from F–N plot To use (5) in the same way as for the planar theory, we might consider plotting Ln(w2 I=OVa2 ) against f ðx; qÞ; but this is not feasible when we do not know the ratio x=Va or q for the experimental conditions. Some information can be obtained from the graph of LnðIÞ against 1=Va : Assuming that a smooth curve can be drawn through the experimental points, the tangent to the curve can be drawn at any point, and its slope S1 and intercept Y1 can be determined. From (5), theoretical values of these are S1 ¼ dLnðIÞ=dVa1 ¼ ðO0 x=O 2w0 x=w 2ÞVa þ S;
ð6Þ
Y1 ¼ LnðIÞ S1 =Va ¼ LnðPÞ þ T=Va S1 =Va ;
ð7Þ
where P ¼ Oc1 f=ðe2 w2 x2 Þ;
53
Independent measurement on the material of one tip characterised in detail showed that the work function was close to 5.0 eV. We have taken both this value and the observed current/voltage relation as given, and have investigated how well the theory above fits these observations. The plot of LnðIÞ against Va1 has a slope S1 of 28953 V and an intercept Y1 of 0.757, at a mean anode voltage of 1625 V. In the case tested, with x in the range 0.1–0.3, iterative solution of (6) provided a consistent set of S; b; a and q for each x: These values enabled P to be calculated by both methods. Values of b; a and F are plotted in Fig. 4(a), to show how these quantities vary with x: For this tip, S1 is about 1.3 times S: The ratio Pexp =Pfe is plotted against x in Fig. 4(b). It can be seen that the experimental value of P approaches the free-electron value most closely where x is about 0.15. This value of x provides best consistency between theory and experiment, if it is assumed that the density of
ð8Þ
T ¼ c2 f3=2 f =bx: The specific value of x for the experimental conditions is not known directly. However, all the quantities except S1 ; Va and Y1 in (6) and (7) can be calculated for given ranges of values of x and q: Some iteration is desirable at this stage to obtain values of a and q consistent with each x: Then for each set of parameters, P can be calculated in two ways. One is from (7), by use of the experimental values of S1 ; Va and Y1 and the calculated T: We denote this value of P by Pexp : Another value of P may be obtained by calculation from (8), and this is denoted by Pfe : We can compare the two calculated values by plotting Pexp =Pfe as a function of x: The calculation of Pfe from (8) assumes that the emitter has the density of states of free-electron theory. If the density of states near the Fermi level is appreciably less, as might occur for a semiconductor emitter, then the value of Pfe given by (8) will be an overestimate. In this case the ratio Pexp =Pfe cannot be expected to approach unity closely.
Fig. 4. (a) Values of b; a and F ; and (b) ratio Pexp =Pfe ; both obtained for tested tip.
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C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
states near the Fermi energy for the tested carbon tip is similar to that for the free-electron model. A further check on consistency is available for the present experiment by comparing the observed value of emitter radius with that predicted at the value of x for maximum Pexp =Pfe : From this value of x; the emitting radius and voltage gradient are calculated as 5.2 nm and 6.5 109 V m1(at the mid-range anode voltage). This value of emitter radius is in good agreement with the observed apex radius of 5 nm. Thus, in this case, the value of x giving agreement between observed and calculated emitter radii also gives closest agreement with the free-electron model. This suggests that the density of states at the Fermi level for the tip tested is within a few per cent of the free-electron value. Although the density of states for the emitter is not known accurately, it is clear that in this case the spherical theory enables consistent sets of parameters to be deduced which correspond to realistic values of emitting radius and surface field. The surface field F deduced in this way from the measured F–N slope and work function is greater than that calculated from the measured tip dimensions by a factor of about 2.6. This could be due to roughness of the emitting surface on a scale smaller than the apex radius, that is on a near-atomic scale. This seems possible in view of the method of deposition of the tip. Other possibilities which do not account for the discrepancies in work function or apex radius include: (a) error in f (for deduction of apex radius); (b) use of planar rather than spherical coordinate system and (c) neglect of variation of emission over the surface of the emitter.
The effective solid angle of emission varies approximately as (applied voltage)0.73 in the field-emitting range. The field strength at a spherical emitting surface is larger than would be estimated by use of planar theory with the conventional F–N plot. Revised expressions have been found for the slope and intercept of a tangent to the graph of Ln(I)Va1 ; but these do not give the work function or field directly. Consistent sets of parameters can be found as functions of x: For the single carbon tip tested, with separately measured work function: Use of planar theory with field calculated from geometry gives a tip radius about 2.6 times that observed, for a wide range of work function. The best fit of spherical theory with experimental data gives a value of tip radius in agreement with the measured value, and surface field about 2.6 times the calculated value. The method of fabrication could have produced atomic-scale surface roughness, leading to a field enhancement of the order of 3 relative to a smooth surface. For this experiment, spherical theory fits the observations much better than planar theory.
7. Conclusions
The transmission factor D for an electron incident on the barrier with energy E is determined by the potential distribution in the barrier region. A substitution for the spatially varying part of the wave function
The rate of change of barrier thickness with applied field is slightly greater for a spherical emitter than for a planar one, for a given field at the surface. For a spherical emitting cap, the image correction factor vðyÞ used in planar theory should be replaced by ½f ðx; qÞ=x in the exponent of the transmission factor, with f ; q and x as defined above.
Acknowledgements The author thanks Prof. L. M. Brown and Prof. U. Valdre" for many stimulating discussions
Appendix A. Transmission factor for potential barrier
c ¼ expðGÞ;
ðA:1Þ
(where G may be complex) allows the time. independent Schrodinger equation to be written as rG rG r2 G ¼ ð2m=_2 ÞðV EÞ:
C.J. Edgcombe / Ultramicroscopy 95 (2003) 49–56
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(proportional to jc22 =c21 j) is approximated by
So, if jr2 Gj5ð2m=_2 ÞjV Ej
ðA:2Þ
then
D ¼ jexpð2 DGÞj Z Eexp ð2ð2mÞ1=2 =_ÞRe ðV Un Þ1=2 dr :
rG rGEð2m=_2 ÞðV EÞ:
ðA:3Þ
The vector =G can be written as the sum of vectors normal and tangential to the local equipotential surface: =G ¼ ð=GÞn þ ð=GÞt
Also, since the direction of =G is nearly radial, r2 GEðð2mÞ1=2 =_r2 Þqðr2 ðV EÞ1=2 Þ=qr: Hence for condition (A.2) to be met at any point in the region where ðV EÞ > 0; it is necessary that
from which
jðq=qrÞðr2 ðV EÞ1=2 Þj5ðð2mÞ1=2 =_Þr2 ðV EÞ:
rG rG ¼ ðrGÞ2n þ ðrGÞ2t : We now define Un and Ut ; two values of energy for a single electron, which sum to the total energy E of the electron and are related to the corresponding =G for that electron and local potential energy V by Un ¼ V ð_2 =2mÞðrGÞ2n ; Ut ¼ ð_2 =2mÞðrGÞ2t : The definition (A.1) shows that the ratio of values of c at positions 1 and 2 is c2 =c1 ¼ eG2 =eG1 ¼ eDG ; where
ðA:4Þ With the experimental tip considered, condition (A.4) is met for ðV 2EÞB5 eV, but not for (V EÞB1 eV, when the two sides of (A.4) become comparable in magnitude. This produces some inaccuracy when D is evaluated by use of (A.3). For greater accuracy where (V E) is small, it would be preferable to improve the approximation for rG or use Airy functions. However, to demonstrate simply the behaviour of the complete barrier, the expression (A.3) will be used here as an approximation for D:
DG ¼ G2 G1 : The change DG over a defined path is given by Z DG ¼ rG ds: So on choosing a path normal to the equipotentials, with coordinate n; through a region of varying V ; DG is given by DG ¼
Z
1=2
ðrGÞn dn ¼ ðð2mÞ
Z =_Þ
ðV Un Þ1=2 dn:
To find the change in amplitude of c caused by the barrier, we take the limits of integration as the bounding values of n at which V 2Un is zero. For the geometry of the emitting tip, the equipotential of the cathode surface is taken to be spherical, while those for greater voltage have slightly different shapes. Within the barrier thickness, the variation of the equipotentials from spherical is small. We therefore approximate by taking the direction of the path as radial, when the transmission function
Appendix B. Approximations for the functions f and w The following approximations have been obtained by fitting simple expressions to calculated values of f ðx; 0Þ and wðx; 0Þ; and by estimating the correction needed to account for non-zero q: The values of q found in practice have been less than 0.1. f ðx; 0Þ; exact solution: (3/2)[arcsin x1=2 =ðx x2 Þ1=2 1 approximation: 2½ð1 xÞ3=2 1 f ðx; qÞ; approximation: f ðx; 0Þ 2q (this implies that @f ðx; qÞ=@x is independent of q which is not exactly correct). w(x,0) is given exactly by ½f ðx; 0Þ þ 2xqf ðx; 0Þ= qx=3x approximation: 1=ð1 xÞ4=3 w ðx; qÞ is given exactly by ½f ðx; qÞ þ 2xqf ðx; qÞ= qx 2qqf ðx; qÞ=qq=3x approximation: wðx; 0Þ þ 4q=3x
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References [1] M. Antognozzi, A. Sentimenti, U. Valdr"e, Microsc. Microanal. Microstruct. 8 (1997) 355. [2] C. Edgcombe, U. Valdr"e, J. Microsc. 203 (2001) 188.
[3] C. Edgcombe, U. Valdr"e, Phil. Mag. B 82 (2002) 1009. [4] R.H. Good, E.W. Muller, . in: S. Flugge . (Ed.), Handbuch der Physik, Berlin: Springer, 1956, pp. 156–231. [5] P. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 2, Academic Press, New York, 1989 (Chapter 44).