Total-energy distribution of field-emitted electrons from a W(100) surface

Surface Science 287/288 (1993) 605-608 North-Holland

surface

science

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Total-energy distribution of field-emitted from a W( 100) surface

electrons

Yasuharu Hirai Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-03, Japan

Received 25 August 1992; accepted for publication 24 November 1992

Total-energy distributions of field-emitted electrons were obtained for a W(100) surface heated to 1800 K. Two humps were observed at electron energies of 0.9 and 2.4 eV above the Fermi level. These humps are reproduced in our calculation of the total-energy distribution, which takes account of the tunneling probability evaluated by matching wave functions at the surface. Our results show that these humps broaden the width of the total-energy distribution at high temperatures.

1. Introduction

2. Experiment

The energy spread of electrons from a W thermal field-emission (TFE) cathode [1,2] is important, because it must be minimized in chromatic-aberration-free electron optics [2,3]. As the temperature of the W field-emission cathode rises, the energy spread, or the FWHM of the total-energy distribution (TED) of the electrons, increases. This increase of energy spread appears in the energy region above the Fermi level. In the literature [2,4], the TED of the W-TFE cathodes has been measured at temperatures below 1600 K. The results [4] have shown good agreement with a free-electron model, but the TED’s were obtained from undefined crystallographic surfaces in the presence of carbon contamination. We have measured the TED of field-emitted electrons from a clean and well-defined crystallographic surface ((100) plane) of a W tip at temperatures up to 2050 K. The main subject of this study is to elucidate band-structure effects of W on the TED above the Fermi level. Typical TED’s show two humps above the Fermi level. These humps are reproduced with our TED calculation which takes account of the band-structure effect.

A retarding-potential energy analyzer of the type originally developed by van Oostrom [5,6] was designed for TED measurements. The analyzer consists of an anode, a lens electrode, a shielding electrode, and a hemispherical collector, as shown in fig. 1. A cathode tip is set 50 mm apart from the anode. A portion of the electrons extracted by applying an anode voltage V,, passes through an aperture at the center of each electrode. The electron beam can be focused near the center of the collector by applying a lens voltage V,, where the center of the collector coincides with that of the shielding electrode. The voltage of the shielding electrode V, is 0 V (ground voltage). The collector is connected to the ground via a picoammeter (V, = 0 V). A retarding voltage is applied to the tip so that the TED can be obtained by differentiating the collector current with the retarding voltage. The apertures at the center of the anode, the lens, and the shielding electrodes are 1.2 mm, 4 mm, and 6 mm in diameter, respectively. Each electrode is equally spaced 3 mm apart. The hemispherical collector has a radius of curvature

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Y Hirai / Total-energy distribution of field-emitted electrons from W(lO0)

606

TIP

r----J TRAJECTORIES

ANODE LENS ELECTRODE SHIELDING ELECTRODE COLLECTOR

,5 mm, Fig. 1. Calculated

electron

trajectories

in the analyzer.

of 10 mm, and its surface is coated with a gold film. A thin fluorescent screen is coated on the anode for field-emission pattern viewing. The analyzer is wrapped by a case made of high permeability metal.

CENTRAL

AXIS

’

Fig. 1 shows the calculated result of the trajectories of electrons whose initial kinetic energy is 5 eV, where V’ = 5000 V, V, = - 155 V, and Vs.= Vc = 0 V. Each trajectory corresponds to electrons emitted from the tip at angles of o”, 0.2”, 0.4”, and 0.6” to the central axis. The result shows that all the electrons passing through the aperture of the anode irradiate the collector surface normally. These characteristics are essential to analyze the total energy of the electrons, and are always obtained with a constant value of VJV, = - 155/5000. The calculated result of the potential distribution is also shown in fig. 2, where we assumed that the potential is given by the Laplace equation. The resolution of the analyzer is estimated to be I 0.2 eV by comparing the energy spreads of measured and calculated TED’s of a W(310) plane, 0.28 and 0.20 eV, respectively, at room temperature. The W tip was prepared by etching a (100) oriented single-crystal wire (0.15 mm in diameter). Then it was spot-welded at the center of a hair-pin curved W wire for resistive heating. The tip and the analyzer were enclosed in an ultra-highvacuum chamber at a base pressure of 2 X 10-s Pa. The tip was flashed several times at 2800 K to clean the surface. We set the anode aperture at the center of the field-emission pattern of the (100) plane. The tip was heated up to 2050 K by a DC current for TED measurement. No magnetic field effect due to the DC current was observed on the field-emission pattern.

1

5000 (V,)

3. Results and discussion -155 (V,) 0 (4) 0 (4)

I

Fig. 2. Calculated potential equipotential cmve shown

5 mm distribution is denoted (volt).

in the analyzer. by its potential

Each value

Experimental results are shown in fig. 3a for a series of electric fields F probing the W(100) plane at a temperature T = 1800 K. The workfunction value 4 is estimated to be 4.65 k 0.05 eV, which is in agreement with a previously-obtained result [6]. The values of F and 4 are determined by combining the value of the slope of the Fowler-Nordheim plot with that of the TED below the Fermi level at room temperature (eqs. (11) and (13) in ref. [61X Each curve shifts to lower energy as the value of F increases due to the narrowing effect of the

(a) W(100) MEASURED

T=1800 K 9 d.65

eV

F (v/A) (1) 0.255 (2) 0.271 (3) 0.295 (4) 0.329

-1

0

1

2

ELECTRON (b) CALCULATED (FREE-ELECTRON

3 ENERGY

4

5

(eV) T=l800 K

MODEL) 9z4.85 eV

1.0

F (v/A) (1) (2) (3) (4) (5) (8)

0.5

0.24 0.28 0.28 0.30 0.32 0.34

0.0 -1

0

I

2

ELECTRON

3 ENERGY

4

5

(eV)

collected simultaneously, from several crystallographic planes, and (3) the fact that the temperature (1570 K) was not high enough to observe humps clearly. TED results calculated by adopting the Sommerfeld free-electron model [7,8] are shown for different values of F in fig. 3b, where we assume the values of T = 1800 K and 4 = 4.65 eV. Each TED curve shows a shift to lower energy as the value of F increases, and shows one peak. Thus, the origin of the humps cannot be explained by the free-electron model. Swanson and Crouser first observed a hump (the Swanson hump) in their TED curves at about 0.37 eV below the Fermi level for a W(100) plane [6]. This hump is also observed in our study. The TED calculation by Nicolaou and Modinos 19,101 showed a pronounced peak corresponding to the hump, and they claimed that the hump could be attributed to a virtual surface state. Their method can reproduce the humps observed in this study, as shown below. The total energy distribution of the fieldemitted electrons N(E) dE is expressed as the number of electrons emitted per unit area per

Fig. 3. (a) Measured TED curves for the W(100) surface. Each curve is normalized to its peak height. The Fermi level is the origin in the abscissa. Humps are indicated by two arrows. (b) Calculated TED curves. Each curve is normalized to its peak height. The Fermi level is the origin in the abscissa.

20000

surface-potential-barrier width. Broad humps appear at energies E = 0.9 and 2.4 eV. These humps can be observed at temperatures above 1600 K because of thermal excitation of the electrons. At 2000 K, humps appear at E = 0.9 and 4.7 eV. The hump at E = 2.4 eV, however, is not prominent because of the overlap of the main peaks. The hump at E = 4.7 eV is due to thermionic emission from the hair-pin curved W wire. Previous results by Gadzuk and Plummer 141,however, do not show any humps. Possible factors that may have smeared out such humps in ref. [4] are: (1) the carbon contamination of the tip surface, (2) the fact that the field-emitted electrons were

5s I-N 23

15000

@ &j Eig

1

FW( 100) CALCULATED a&.21 A T=l800 K 4=4.65 eV F4.29 VIA

10000

CCx a E

5000

0 -1

0

1

2

3

4

5

ELECTRON ENERGY (eV) Fig. 4. Calculated TED curves for the WUOO) surface. The Fermi level is the origin in the abscissa.

608

Y. Hirai / Total-energy distribution of field-emitted electrons from W(100)

unit time with energy between This is given by [9,10] N(E)

E and E + dE.

dE dk E, k,,)-

(2r)3

where f(E) is the Fermi-Dirac distribution function. k = (k I, k,,) is the wave vector of electrons inside the metal, where k I is the component normal to the W(100) plane and k,, = (k,, kZ) is the component parallel to the plane. k,, lies within the first Brillouin zone of the two-dimensional reciprocal lattice. T(k I ; E, k,,) is the tunneling probability for an electron, and it is calculated by matching the wave function outside the metal to a superposition of Bloch waves inside the metal. Bloch wave functions are given by solving the eigenvalue equation ill]. The lattice constant of W, a, is expressed as a = 3.16[1 + 1.0 x lo-‘(T300)] A [12]. At 1800 K, the value of a equals 3.21 A. For a given energy E we have calculated the Bloch waves and tunneling probability for a number of k,, values within the two-dimensional Brillouin zone. The points are obtained by dividing the Bri!ouin zone into steps of dk,, = dk, = (2-rr/a)/50 A-‘. Fig. 4 shows the result of our numerical calculation of, the TED of the W(100) surface for a = 2.21 A, T = 1800 K, 4 = 4.65 eV, and F = 0.29 V/A. The calculation gives pronounced peaks in the energy region where the humps appear experimentally in fig. 3a. The calculated peak intensity at E = 2.6 eV is 1.74 X lo4 electrons/eV/A2/s. This value can be compared to the moeasured peak intensity of TED at F = 0.271 V/A in fig. 3a. We estimate the measured peak intensity to be 2.38 x lo4 electrons/eV/A2/s from the total current density (100 A/cm*). In fig. 3a, the hump at E = 2.4 eV is broader than the hump at E = 0.9 eV. This might be explained by the multiple humps at E = 2.3-2.8 eV in fig. 4.

4. Conclusions A retarding-potential analyzer of the type designed by van Oostrom was used for TED measurements. The observed TED curves of the W(100) surface at 1800 K show humps at 0.9 and 2.4 eV above the Fermi level. These humps deviate from the curves calculated by the free-electron model. The observed humps are reproduced by a calculation which takes account of the tunneling probability of the Bloch waves inside the metal.

Acknowledgements

We gratefully acknowledge Dr. Shigeyuki Hosoki, Dr. Hideo Todokoro, Dr. Shigehiko Yamamoto, Dr. Hiroshi Okano, Dr. Kenji Utagawa, and Dr. Izumi Waki for their helpful discussions.

References [l] A.B. El-Kareh, J.C. Wolfe and J.E. Wolfe, J. Appl. Phys. 48 (1977) 4749. [Z] L.W. Swanson, J. Vat. Sci. Technol. 12 (1975) 1228. [3] A.V. Crewe, Q. Rev. Biophys. 3 (1970) 137. [4] J.W. Gadzuk and E.W. Plummer, Phys. Rev. B 3 (1971) 2125. [5] A.G.J. van Oostrom, Philips Res. Rep. Suppl. 1 (1966) 49. [6] L.W. Swanson and L.C. Crouser, Phys. Rev. 163 (1967) 622. [7] R.D. Young, Phys. Rev. 113 (1959) 110. [S] S.C. Miller, Jr. and R.H. Good, Jr., Phys. Rev. 91 (1953) 174. [9] N. Nicolaou and A. Modinos, Phys. Rev. B 11 (1975) 3687. [lo] A. Modinos and N. Nicolaou, Phys. Rev. B 13 (1976) 1536. [ll] J.B. Pendry, J. Phys. C 4 (1971) 2501. [12] N.E. Christensen and B. Feuerbacher, Phys. Rev. B 10 (1974) 2349.