ARTICLE IN PRESS

Ultramicroscopy 109 (2009) 403–412

Contents lists available at ScienceDirect

Ultramicroscopy

0304-39

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ultramic

Improving the energy spread and brightness of thermal-field (Schottky)emitters with PHAST—PHoto Assisted Schottky Tip

Ben Cook �, Merijn Bronsgeest, Kees Hagen, Pieter Kruit

Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

a r t i c l e i n f o

PACS:

29.25.Bx

85.60.Ha

79.60.�i

70.40.+z

Keywords:

Photo

Photoemission

Photofield

Schottky

Electron

Emission

Cathode

Source

91/$ - see front matter & 2008 Elsevier B.V. A

016/j.ultramic.2008.11.024

esponding author.

ail address: b.j.cook@tudelft.nl (B. Cook).

a b s t r a c t

Using a relatively simple model of photoemission we derive an expression for the reduced on axis

brightness of a thermal-photofield emitter. We then show that it is theoretically possible to reduce the

energy spread of a Schottky (thermal field) emitter whilst increasing the reduced brightness. This can be

achieved by the illumination of the tip with a high intensity laser light. We call the source

PHAST—PHoto Assisted Schottky Tip. We find that due to the strong E-fields applied PHAST may operate

at photon energies below the (Schottky reduced) work function. Thus removing the need for UV lasers,

we will show that it is in fact preferable to work in the red, or in the green. The necessary laser

intensities probably limit the application to pulsed operation.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The Schottky electron source (tungsten needle thermal-fieldemitter with a layer of ZrOx, see Fig. 1) is used for many SEMs,TEMs, Auger spectrometers and semiconductor inspection tools.Electron emission arises due to a combination of thermal and fieldeffects. The ZrOx lowers the work function of the tungsten toabout 2.9 eV. Schottky sources are normally operated at fieldsbetween 0.4 and 1.0 V/nm, provided by an extractor plate in frontof the tip. A temperature of approximately 1800 K is obtained fromresistive heating. Typical parameters are an energy spreadbetween 0.3 and 0.8 eV full width 50 (FW50) and an experimen-tally measured reduced brightness of 107 A=m2 sr V [1]. Thebrightness is a key concept in optics, and it measures the currentdensity per solid angle. Unfortunately the brightness is notinvariant to an accelerating/decelerating voltage and so we usethe brightness per volt; this quantity is called the reducedbrightness. Energy spread and reduced brightness are limitingfactors in many applications. It has been shown [2,3] that formany situations the reduced brightness and energy spreaduniquely determine the amount of current that can be focusedinto a spot of a particular size.

ll rights reserved.

We intend to control energy spread and/or increase thereduced brightness by illuminating a Schottky tip with a laser.Thus making a ‘‘PHoto Assisted Schottky Tip’’—PHAST.

In the rest of this paper we will first bring togetherShimoyama’s work [4] and the Fowler–Dubridge method [5–7]to derive equations for the reduced brightness of a photoemitter.This will be the main topic of Section 2.1 but it will also includecurrent density calculations which are useful when comparingtheory to experiment. Section 2.2 will give an equation to find theenergy spread. In Section 2.3 we will discuss the Fowler–Dubridgemethod and what we call Fowler’s approximation. Section 2.4gives a short overview on how we deal with the transmissioncoefficient, and Section 2.5 is on scattering in metals. Section 3.1will include a comparison of our theory to experimental andanalytical results. In Section 3.2 we will use our model to showthat PHAST can improve the reduced brightness and energyspread of a Schottky emitter. Finally we will have a discussionabout the usefulness of PHAST and make some conclusions.

2. Introduction to PHAST

The basic concepts of PHAST emission are schematically shownin Fig. 2. For most cathodes electrons can escape by any of thethree basic processes: thermal emission over the top of the barrier(see Fig. 2); field emission by tunneling through the barrier; or

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B. Cook et al. / Ultramicroscopy 109 (2009) 403–412404

photoemission, where an electron interacts directly with a photonand escapes over the barrier. In the present situation we have astrong field, a high temperature and a focused laser beam appliedto the tungsten tip as shown in Fig. 1. This means that we can haveemission by combination of any of the three previously mentionedinputs.

(1) The strong field will act to lower the barrier (see Fig. 2); thisis the Schottky effect [8]. The barrier shape changes as shown inFig. 2 for a field strength of 1 GV/m. If the field is strong enoughthe barrier will become thin enough for electrons to tunnelthrough.

(2) The high temperature changes the energy distribution ofthe electrons giving it a long tail at higher energies. Some of theelectrons in the tail will have an energy above the barrier heightand can then escape. This may be assisted by the Schottky effect,which will reduce the temperature required for a particularcurrent. Thermally excited electrons may also tunnel through thetop of the barrier where it is thin enough.

Fig. 2. Schematic description of the emission processes contributing to PHAST. The dash

of a strong E-field; this allows tunneling to occur. In the case of a Schottky emitter the tu

significant tunneling from around m (the electron chemical potential). With PHAST we h

an energy level hn higher. These excited electrons have the possibility to escape either

excited and are subsequently excited by a photon, and escape, then they become therma

shown with the line marked with *.

Fig. 1. Electron microscope picture of a standard Schottky tip, the lump is the ZrOx

reservoir. The legs are for heating.

(3) Finally, the focused laser beam (ignoring any thermalaspects) will promote electrons to higher energy levels. Theelectrons can then do one of the three things: (i) escape directly;(ii) escape in combination with their prior thermal energy; or(iii) they may be photoexcited and then tunnel if there is sufficientfield.

We want to use the three inputs to maximize the output ofPHAST for the highest brightness and lowest energy spread.

We will need equations to describe the reduced brightness andenergy spread. We will also want equations for the quantumefficiency/current density so that we might compare our model toexperimental results. For thermal-field emission these are welldocumented in textbooks and in the literature, see for example[9,10]. Photoemission literature is often based on the Fowler–Du-bridge model [5,6]. With this we may calculate energy spreads,and currents or quantum efficiencies and even account for multi-photon excitation [11]. However, up to now the Fowler–Dubridgemodel does not include a method of finding the brightness. This iswhat we will now do.

2.1. Brightness

To derive an expression for the reduced brightness from aphotoemission source we will start by looking at the work ofShimoyama [4]. We will follow part of his derivation of brightnessfor a thermal-field emitter. Then we will combine Shimoyama’swork with Jensen’s work on photo (field) cathodes (see forexample [7]).

According to Shimoyama [4] to derive a complete (at everypoint in space) expression for the brightness, we must ray tracethe emitted electrons. The full expression for brightness is shownin Eq. (1) and illustrated in Fig. 3:

Bðr; yÞ ¼d2Iðr; yÞ

dOdS cos y(1)

Here r and y refer to coordinates on a cylindrical coordinatesystem, I is the current, dS a cathode surface element, dO is thesolid angle element, and y the angle normal to dS. For some cases,analytical solutions can be found in Eq. (1), but not in general, and

ed and �marked lines show how the barrier shape changes with the introduction

nneling is mainly from thermal electrons as the field is not strong enough to allow

ave the addition of photons which excite electrons from lower levels to (if available)

directly or by tunneling through the barrier. If the electrons are already thermally

l photo (field) emitted electrons. The distribution of emitted electrons for PHAST is

ARTICLE IN PRESS

dΩ

θ

r

P

dS

Fig. 3. An Illustration to help with the definition of optical brightness.

B. Cook et al. / Ultramicroscopy 109 (2009) 403–412 405

we therefore seek alternatives. In electron optics the maximumbrightness is often used, or using the terminology of Shimoyama,the axial brightness B0—the brightness on the axis in the axialdirection. So r and y become zero and we assume a uniformcurrent distribution and replace I with J the current density

B0 ¼dJ

dO(2)

But we would like to have the reduced brightness, i.e. thebrightness per volt B0=V

B00 ¼dJ

dOV(3)

We will now find the on axis (maximum) reduced brightness ofPHAST, starting from an expression for the non-reduced bright-ness; Eq. (15) in [4] which is in general valid

B0 ¼e2

pF�Z 1�1

NðEn; Et ¼ 0ÞdEn (4)

where F� is the relativistically corrected accelerating voltage,NðEn; Et ¼ 0Þ is the energy distribution at zero tangential energy(Et). En is the energy of the electron normal to the cathode surface.This may also be written as

B0 ¼e2

pF�NðEt ¼ 0Þ (5)

where NðEt ¼ 0Þ is the emitted energy distribution at zerotangential energy. The tangential energy distribution of theemitted electrons is simply

NðEtÞ ¼

Z 1�1

NðEn; EtÞdEn (6)

We now skip to Section 5 in Shimoyama [4] and Eq. (24) wherethe expression for the emitted tangential energy distribution NðEtÞ

is given for thermal-field emission as

NðEtÞdEt ¼ dEt

Z 1�1

PðEn; EtÞDðEnÞdEn (7)

where

PðEn; EtÞ ¼A

ek21þ exp

En þ Et þfw

kT

� �� ��1

(8)

Here DðEnÞ is the transmission factor (an analytical approximationis given by Shimoyama and a numerical method is described herein Section 2.4). P is a 2D thermal supply function, withA ¼ 4pmek2=h3, e is the elementary positive charge, m is theelectron mass in free space, k is Boltzmann’s constant, h is Planck’sconstant, T is the thermodynamic temperature and fw is theenergy gap between the zero energy point and the Fermi energy.Shimoyama defines the zero energy point as the energy of anelectron at an infinite distance from the metal with zero field. Inthe energy scale shown in Fig. 2 this would be the barrier height h.

In Fig. 2 fw is h� m where m is the chemical potential (we neglectthe temperature dependence of m and assume m ¼ mðT ¼ 0 KÞ).Now with our expression for P substituted in Eq. (5) we may writedown an expression for the reduced on axis brightness of athermal-field emitter as

B00tf ¼e2

p

Z 10

PðEn; Et ¼ 0ÞDðEnÞdEn (9)

We are now ready to combine the expression of Shimoyama forthe axial brightness with Jensen’s work on photocathodes. Wewant Eq. (9) in the presence of photons.

Jensen works with the same energy scale as in Fig. 2 so wemust convert Shimoyama’s to Fig. 2 so we can have an equivalentexpression for the reduced on axis brightness. Beginning with the2D thermal supply function which becomes

Q ðEn; EtÞ ¼A

ek21þ exp

En þ Et � mkT

� �� ��1

(10)

where for clarity we have used the new variable Q. The newtransmission coefficient becomes D0ðEnÞ ¼ DðEn � hÞ. Now Eq. (7)again using a new variable M instead of N becomes

MðEtÞdEt ¼ dEt

Z 10

Q ðEn; EtÞD0ðEnÞdEn (11)

Now Eq. (9) becomes

B00tf ¼e2

p

Z 10

MðEt ¼ 0ÞdEt ¼e2

p

Z 10

Q ðEn; Et ¼ 0ÞD0ðEnÞdEn (12)

We may now note that if we carry out the integration of MðEtÞ inEq. (11) with respect to Et over all energies we would end up withan expression for the flux of emitted thermal-field electrons. Moreprecisely for flux we mean the number of thermal-field emittedelectrons per second per meter squared. Multiplying this flux by e

gives the current density:

Jtf ¼ e

Z 10

MðEtÞdEt ¼ e

Z 10

Z 10

Q ðEn; EtÞD0ðEnÞdEn dEt (13)

Eq. (5) for the reduced on axis brightness can be written in theform used by Fransen [12],

B00 ¼e

pdJ

dEt

����Et¼0

(14)

Now linking to Jensen we find that integrating Q over Et we willend up with (excluding some pre-factors which will in any casecancel out) what Jensen refers to as the 1D supply function

FðEnÞ ¼A

kT ln 1þ exp

ðm� EnÞ

kT

� �� �(15)

In [13] he writes this as

f ðEÞ ¼mpb_2

ln 1þ expðm� EðkÞbÞ� �

(16)

and in Eq. (47) of [ 14] in a form closer to that derived here. Jensenuses b to represent 1=kT and E (as a function of the quantum wavenumber k, not Boltzmann’s constant) is our En.

We now need to combine Shimoyama and Jensen. We want tofind the brightness in the presence of photons. So we will need anexpression for MðEt ¼ 0Þ (the tangential energy spread at zerotangential energy) when illuminated by a laser. To include laserillumination we need some idea about the quantum efficiency,which relates laser intensity at the cathode to the emitted currentdensity in the following way:

J ¼ð1� RÞIe

hn g (17)

where R is the reflectivity of the cathode, I is the laser intensity(not to be confused with the current of Eq. (1)), and g is the

ARTICLE IN PRESS

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

Photon Energy (eV)

B’ 0P

HA

ST

/ JP

HA

ST

(SR

−1V

−1)

e/πkBTB’0 T=0 /JT=0B’0PHAST/J’0PHAST

WorkFunction Setat3eV

Fig. 4. Reduced on axis brightness (B00PHAST ) divided by total current density (JPHAST )

plotted against photon energy (red with � markers) and for comparison e=pkT

which if multiplied by Jtf gives the reduced on axis brightness of a (low field)

thermal emitter. The two lines do not quite match because of numerical

inaccuracies and the approximation of Eq. (26). The third line (blue with circle

markers) is for cold photoemission and so we see this form dominates for larger

photoenergies. For low photon energies B00T¼0=J0T¼0 tends toward infinity, but for

any situation where a reasonable amount of current is produced it is below the line

of e=pkT. T ¼ 1800 K.

B. Cook et al. / Ultramicroscopy 109 (2009) 403–412406

quantum efficiency. In [13] Jensen tells us that based on theFowler–Dubridge method [5,6] (see Section 2.3 for a discussion ofthis) the quantum efficiency of a photocathode is the numberof emitted electrons per absorbed photon and he gives this as(Eq. (4) of [13])

QE ¼

RW0

R10 f ðEÞD0ðEþ hnÞdE

� �exp �

rr0

� �2 !

rdr

RW0

R10 f ðEÞdE

� �exp �

rr0

� �2 !

rdr(18)

W, r, and r0 are the factors introduced from the geometry of thecathode and the distribution of the laser beam. In our simplemodel we work on the assumption of a semi-infinite planecathode, and a uniform laser beam, so the integrals over dr and allterms with r can be replaced by current densities and laserintensities. This simplification is valid for thermal-field emittersbecause in microscopy we aperture the beam, and thus see only afraction (10’s of nm) of the emission area. For a first approxima-tion of a photoemitter this simplification tells us roughly whatlaser powers we will need. Again due to aperturing it is probablyalso reasonable.

We are almost ready to use this expression for the quantumefficiency to calculate our total current for PHAST. First we need toinclude the scattering of the electron on the way to the surface ofthe cathode. We do this in a slightly different and more simplemanner than Jensen who discusses his method in [7]. We includethe scattering term SðEnÞ in the upper integral of Eq. (18), so wedefine the quantum efficiency with g and write

g ¼R1

0 SðEnÞFðEnÞD0ðEn þ hnÞdEnR1

0 FðEnÞdEn(19)

The scattering term S is discussed later in Section 2.5. Jensen alsostates that photoemission current simply adds to the thermal-field current [15]. So now we may write an expression for the totalcurrent from PHAST using FðEnÞ instead of f ðEnÞ,

JPHAST ¼ Jtf þIð1� RðhnÞÞe

hn

R10 SðEnÞFðEnÞD

0ðEn þ hnÞdEnR1

0 FðEnÞdEn(20)

We would like to make the form of the equation more clear, so wewill rewrite it with the substitution of the variable k

JPHAST ¼ Jtf þ kZ 1

0SðEnÞFðEnÞD

0ðEn þ hnÞdEn (21)

k ¼ Ið1� RðhnÞÞehn

1R10 FðEnÞdEn

(22)

We may think of k as a correction or scaling factor forphotoemission where the integral in the denominator is themaximum current available from all the electrons in the metal,and the remaining terms represent the total charge which thephotons have interacted with or the number of photons absorbedby the metal multiplied by eZ. With this in mind it follows fromEq. (13) that

JPHAST ¼ Jtf þ kZ 1

0

Z 10

SðEnÞQ ðEn; EtÞD0ðEn þ hnÞdEn dEt (23)

and so the reduced axial photobrightness B00ph ¼ ðe=pÞdJph=dEtjEt¼0

is

B00ph ¼e

pkZ 1

0SðEnÞQ ðEn; Et ¼ 0ÞD0ðEn þ hnÞdEn (24)

and

B00PHAST ¼ B00ph þ B00tf (25)

The reduced on axis brightness is widely used in microscopy forthermal and field emission, and so the expression works well as asource comparator.

Let us see how our PHAST emitter might compare with astandard Schottky tip. If we define brightness efficiency as B00=J,then we find that a photoemitter is less efficient than a thermal-field emitter operating at 1800 K (the manufacturers recom-mended operating temperature for our Schottky tip), and this isillustrated in Fig. 4. We see that B00PHAST=JPHAST decreases withincreasing photon energy, and for sub-threshold photon energythe reduced on axis brightness is equivalent to a thermal-fieldemitter B00 ¼ eJ=pkT [4] (if tunneling can be ignored). This resultand an expression for the brightness with zero field and zerotemperature will be derived. We start with the sub-thresholdphotoemission (with zero field); here electrons escape only by acombination of thermal energy and photoexcitement. We willbegin by adopting Shimoyama’s notation, and energy scale (whichwill prove easier here). With Eþ fw large compared to kT, we maywrite Eq. (8) as

P� ¼A

ek2exp �

En þ Et þfw

kT

� �(26)

Now if we neglect quantum mechanical effects, assume anaveraged scattering factor S, and a step-like barrier function ofheight h ¼ fW þ m, then for sub-threshold zero field photoemis-sion, we can write

J� ¼ kS

Z 10

Z 10

A

k2exp �

En þ Et þfw � hnkT

� �dEn dEt (27)

where we have moved the effect of the photon to act directly onthe work function fw. This makes the derivation more clearlyfollow that of the thermal-field derivation,

J� ¼ kSAT2 exp �fw � hn

kT

� �(28)

and the reduced on axis brightness B0�0 will be with Et ¼ 0

B0�0 ¼ kS

Z 10

e

pA

k2exp �

En þfw � hnkT

� �dEn (29)

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B. Cook et al. / Ultramicroscopy 109 (2009) 403–412 407

B0�0 ¼ kSe

pA

kT

k2exp �

fw � hnkT

� �(30)

finally we see that

B0�0 ¼eJ�

ðpkTÞ(31)

Now we will discuss the zero temperature, zero field photoemis-sion case. We will start with the expression for the currentdensity, Eq. (20). In this case Jtf ¼ 0 and again we use S. We canthen write

JT¼0 ¼Ið1� RðhnÞÞSe

hn

" # R mðfWþm�hnÞðm� EnÞdEnR m

0 ðm� EnÞdEn

(32)

Integrating gives us

JT¼0 ¼Ið1� RðhnÞÞSe

hn

" #hn2 � 2hnfW þf2

W

m2

(33)

Now, recalling eq. (14), for the reduced on axis brightness we maywrite

B00T¼0 ¼Ið1� RðhnÞÞSe

hn

" #eR mðfWþm�hnÞð1ÞdEn

pR m

0 ðm� EnÞdEn

(34)

Integrating once more leads to

B00T¼0 ¼Ið1� RðhnÞÞSe

hn

" #2eðhn�fW Þ

pm2(35)

Eqs. (31) and (35)are shown in Fig. 4 and compared to a numericalcalculation with zero field, at a temperature of 1800 K. We see thatfor photon energies less than fW the reduced on axis brightnessfollows that of a thermal emitter, and for photon energiessufficiently over threshold it follows Eq. (35).

2.2. Energy spread

We have still not considered the energy spread. This can becalculated by the consideration of a total energy distribution, andthe end results match closely with Dubridge’s [6] exact expressionfor the distribution of photoexcited electrons using a step barrierfunction. We will simply quote the result because the derivation isso similar to the one for the brightness and well documented forthermal-field emission; see for example [10]

dJtf

dE¼ k A

k2SðEÞ 1þ exp

E� mkT

� �� ��1 Z E

0D0ðEnÞdEn (36)

and for photoemission this will be

dJph

dE¼

Ið1� RðhnÞÞehn k A

k2SðEÞ 1þ exp

E� mkT

� �� ��1 Z E

0D0ðEn þ hnÞdEn

(37)

where we use E to describe the total energy of the electron. Theproblem with this is that the location of the emitted electrons willnot be correct in comparison with the thermal-field electrons. Thesolution to this is to shift the supply function (instead of shiftingthe transmission coefficient to lower energies) to higher energyand so we get

dJph

dE¼ k A

k2SðEÞ 1þ exp

E� ðmþ hnÞkT

� �� ��1 Z E

0D0ðEnÞdEn (38)

finally the two terms can be added

dJPHAST

dE¼

dJtf

dEþ

dJph

dE(39)

2.3. The Fowler–Dubridge model and Fowler’s approximation

Looking at Eq. (20) we see that the effect of photoexcitationis to shift the transmission function to lower energy by anamount hn. The transmission function is dependent on En. In otherwords all of the photon’s energy goes into increasing the normalenergy of an electron. This is what we shall call Fowler’sapproximation.

A better theory would be for the photon’s energy to add to thetotal energy of the electron, E0 ¼ Eþ hn, but we are not aware ofthis leading to an easily manipulated form. Fowler was the first tocreate a successful theory of photoemission [5] with an analyticalform that could be easily applied. He was interested in a theorywhich could deal with the ‘thermal tails’ which made determiningwork functions from photoemission data difficult.

Fowler suggested an approximation for near threshold emis-sion where E0 equals the barrier height. Allow the photon’s energyto add only to the En component, E0n ¼ En þ hn. This is reasonable,since E0 ¼ E0t þ E0n, so at threshold E0t must be zero to allowemission. Fowler did not include field emission in his originalpaper, so he used a step-like barrier, D0ðEn þ hnÞ ¼ 0 for En belowthe barrier and D0ðEn þ hnÞ ¼ 1 for En above the barrier which canbe easily taken account of with integration limits. Jensen takesaccount of the photon energy within a transmission coefficient sowe can include field emission. Fowler’s approximation is validatedand used countless times in the literature for example [5,6,16–19].As well as in this paper, see Figs. 8–10.

It is worth noting that our expression for the reduced on axisbrightness does not suffer from Fowler’s approximation. Et mustbe zero, so E0t must also be zero, and E0 ¼ E0n ¼ En þ hn.

2.4. Transmission factor DðEnÞ

To find DðEnÞ we assume that a uniform electric field F isapplied at the cathode surface and then consider the potential ofan image charge to find the barrier potential VðzÞ; the result isgiven in Eq. (40) (see [9])

VðzÞ ¼ mþ fW �e2

16p�0z� Fze (40)

where �0 is the permittivity of free space. Eq. (40) is takenonly for positive values of VðzÞ and is zero everywhere else,see Fig. 5.

Now we may solve the Schrodinger equation as on both sidesat z1 and z2 there is zero potential. We end up with a wavefunction c made up of three waves/components:

cðzÞ ¼ C1 exp ikzzþ C2 expð�ikzzÞ for zoz1

cðzÞ ¼ C3 expðikzzÞ for z4z2 (41)

where C1 and C2 are the amplitudes of the electron wavestraveling left and traveling right for zoz1, and C3 is the amplitudeof the electron wave in the vacuum at zero potential. Using afourth order Runge–Kutta method we step through the barrierfrom right to left and solve to obtain

DðEnÞ ¼C3

C1

��������2

(42)

Now we may find Jtf at a range of fields and temperatures. Wehave used a chemical potential of 10 eV. Jtf has been plotted inFig. 6 where we can see how it compares to its analyticalcounterparts, and in Fig. 7 where we can see how it compares toexperimental data. However, to calculate the photocurrent Jph wemust first consider the scattering.

ARTICLE IN PRESS

Fig. 5. Potential energy barrier for calculating transmission coefficient DðEnÞ

calculated from Eq. (40). Zero is taken as the lowest energy level in the conduction

band of the metal. The dotted line shows the true potential which continues to

decline toward the anode.

Fig. 6. Current density for thermal-field emission. The temperature is 1800 K and

the work function is 2.7 eV, and the field is varied from 0.4–2 V/nm. + line shows

the extended Schottky current density. The squares line is numerically calculated

using the model described in Section 2. Circles are for the Schottky equation. Note

that there is a large difference (the vertical axis is log scale) between the three

lines at high fields. In the equation fW �DfW is the Schottky lowered work

function and q is a function of field and temperature which deals with the

tunneling [12].

Fig. 7. Current density derived from experimental data (dotted line diamond

markers) and numerical (solid line) results for a standard Schottky emitter.

Temperature is 1800 K, work function is 2.95 eV, other data are taken from [23].

Note there is good agreement between the numerical and the experimental data

especially at higher fields where analytical methods tend to fail.

B. Cook et al. / Ultramicroscopy 109 (2009) 403–412408

2.5. Scattering

The previous section would be sufficient for calculatingemission from a metal if the photoelectric effect was purely asurface phenomenon, which would mean no scattering. Since thisis not the case, we must include a scattering term which is nowdiscussed.

Although there is plenty of data on mean free paths and escapedepths of electrons for the energy range 50 eV and above, below50 eV the data are sparse. Thus we use tabulated values of the

thermal conductivity and infer the mean free path (le) from this[20]. In our case we are mainly interested in electrons close tothreshold and thus we again make the approximation that theelectrons only travel the perpendicular distance to the surface. Wealso make the assumption that one collision is enough to preventthe escape which is probably true at such low photon energies.The probability of an electron being transported to the surface isgoverned by the following exponential decay law expð�z=leðEnÞÞ,where z is the distance to the surface in the normal direction. Thismust also be related to the absorption depth d of the light (againgoverned by an exponential decay law). d is a function of hn andvalues are taken from the CRC handbook [21]. Combining theseparameters we obtain Eq. (43) which describes the probability of aphotoexcited electron to scatter

SðEnÞ ¼

R10 exp �

z

dðhnÞ

� �exp �

z

leðEnÞ

� �dz

R10 exp �

z

dðhnÞ

� �dz

¼1

1þdðhnÞleðEnÞ

(43)

where z is the distance from the surface of the cathode, le is theelectron mean free path and d is the absorption depth of the light.When calculating the total energy distribution the approximationbecomes SðEnÞ ¼ SðEÞ because we do not have a term for En.

Eq. (43) is not new, Spicer has used this in [22]. What isperhaps different is the simple use of thermal conductivities incombination with the approximation of only normal travelingelectrons. Because of this, it is a very crude approximation. In factthese values are only really valid around the Fermi energy so ourextrapolation to other values is quite unjustified. Luckily theresults seem to fit the previous experimental data well, seeFigs. 8–10; so we are for the time being satisfied with this method.

3. Results

3.1. Comparison with existing data

We must check that our model is correct; to do this we firstcompare our results with the analytical equations of thermal-field

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B. Cook et al. / Ultramicroscopy 109 (2009) 403–412 409

emission, and with experimental results. Also we will check thephotoefficiency (number of emitted electrons/number of incidentphotons) against the experimental data.

1.5 2 2.5 30

1

2

3

4

5

6

Work F

Pho

to E

ffici

ency

x 10−4

Fig. 10. Photoefficiency of cesiated tungsten dispenser cathode plotted against work fun

stars show experimental data. The laser wavelength used was 532 nm (2.34 eV), assum

Fig. 8. Photoefficiency vs wavelength of illuminating light. Experimental data from

[24] for gold (squares) and silver (diamonds) plotted with our PHAST simulation

data (dotted lines).

Fig. 9. Photoefficiency vs wavelength of illuminating light. Experimental data from

[24] for potassium (circles after ion cleaning, triangles after vacuum distillation,

diamond after out gassing), plotted with our PHAST simulation data (dashed lines).

Fig. 6 compares our results for thermal-field emission with theSchottky and extended Schottky equations (44) and (45)

JS ¼ ðAðkBTÞ2Þ exp �ðfW �DfW Þ

kBT

� �(44)

JES ¼ Js

pq

ðsinðpqÞÞ(45)

where fW �DfW is the Schottky lowered work function and q is afunction of field and temperature which deals with the tunneling.Eqs. (44) and (45) are derived and compared with experimentalvalues in [12]. Essentially the Schottky equation deals with thelowering of the emission barrier by the high applied field; thisgoes into DfW but does not include tunneling. The extendedSchottky model was developed to include tunneling, whichbecomes significant at high field and temperature.

At low fields all three lines agree, but diverge at higher fieldswhere electron tunneling becomes significant, and is eitheroverestimated (extended Schottky) or neglected (Schottky).

Now if we compare our numerical results with some experi-mental data (Fig. 7) for Schottky emission we find excellentresults. No fitting factor is used, and constants such as workfunction are the same as reported in [23]. There is some deviationbetween the two lines at lower fields, and this might be explainedby the realignment of the ZrO layer on the tip, or tip facet radiusgrowth at low field and high temperature.

These two results show that it is extremely worthwhile tonumerically integrate the Schrodinger equation for high tempera-ture emitters when electron tunneling becomes important.

We look now at actual photoemission photoefficiencies andcompare with the results presented in [24]—see Figs. 8 and 9 . Thesimulation is fitted for zero field and room temperature. Weassume these were the experimental details. It is clear that withpotassium there is a significant difference depending on thecleaning method used, as from the figure we see this makes a bigdifference to the work function or more accurately the thresholdwavelength for emission (for the simulation we used the workfunction listed in the CRC handbook [21]). We see immediately theproblem of obtaining reliable measurements. Three different setsof data are obtained for potassium, none of which appear to agreewith the work functions presented in the CRC handbook andverified in other locations. Thus it is clear that small surfaceimperfections dramatically change the photoefficiency, and we

3.5 4 4.5 5unction eV

Expermental DataPHAST Simulation

ction calculated from amount of cesiation. Solid line represents PHAST data and the

ed temperature of 300 K and zero or very low electric field.

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are happy with the results here as they have the same shape as theexperimental data and in the case of potassium sit in the ‘middle’of the experimental data. We see with gold and silver that atshorter wavelengths (higher photon energies) we overestimatethe photoefficiency; this is probably because of the approximationof Fowler (see Section 2) but we still obtain the shape, and closerto threshold where the majority of the electron’s energy is in thenormal direction we match well considering the uncertainty insurface conditions. We conclude that the simulation matchesreasonably well the experimental data. The only fitting was toinfer the work function/barrier height from the experimentaldata; all other data were obtained from the CRC handbook [21,25].We can also examine how our results compare with the recentwork of Moody who is developing cesiated photocathodes for usein free electron lasers and accelerators. This should be closer tothe PHAST situation because of the work function lowering ZrOx

1060.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Ene

rgy

Spr

ead

(eV

)

T = 1500K Field 0.4V/nm−2V/nmT = 1650K Field 0.4V/nm−2V/nmT = 1800K Field 0.4V/nm−2V/nm1V/nm T = 1500K 10TW/m2 Laser Density hv = 01V/nm T = 1500K 100TW/m2 Laser Density hv =0.5V/nm T = 1750K 100TW/m2 Laser Density hv

Increasing Field

Increasing Photon Energy

SchottkyRegime

107

Reduced Axial Br108

Fig. 11. Energy spread (FW50) and reduced on axis brightness. Thermal-field emission w

PHAST emission with increasing photon energy from left to right (solid lines), marke

traditional thermal-field sources, whereas the regime in the bottom right (PHAST Regim

is given in the key for this graph.

11.80 12.00 12.20 12.40 12.60 12.80

0.5

1

1.5

2

2.5

3

3.5

4

Energy Sp

dJ/d

E (A

/eV

)

FWHM =

F

FWHM = 0.37eV

FWHM = 0.38eV

x 1027

Fig. 12. Distribution of emitted electrons from bottom to top: (1) pure thermal-field

(497 nm) photons, (5) 3 eV (412 nm) photons. We show here that with red photons we ar

0.38 to 0.37 eV (FWHM) compared with the pure thermal-field case. For shorter wavel

layer on the tip. In this case the fit is excellent (see Fig. 10). Thesituation here is more complex having to calculate how theamount of cesiation relates to work function. This was donegraphically by comparing with Fig. 17 in [26]. We believe that aSchottky tip should behave in a similar way, and that it has aconstant work function lowering effect from the coating of ZrOx,assuming we maintain a steady temperature.

3.2. Simulation results for PHAST

What we want with PHAST is a bright monochromatic electronsource. We believe that this is possible by illuminating a Schottkycathode with a focused laser beam. By combining these twoqualities (reduced on axis brightness and energy spread) on onegraph (Fig. 11), we see that the upper left-hand section of the

−3eV 0−3eV = 0−3eV

PHASTRegime

ightness (A/(m2SRV))109 1010 1011

ith increasing field from left to right (dotted lines), marker spacing of 0.1 V/nm, and

r spacing of 0.2 eV. The region in the upper left denotes the regime available to

e) represents what is available to PHAST. A description of what each line represents

0 13.0 13.20 13.40 13.60 13.80 14.00read in eV

No Photons2.0eV Photons2.25ev Photons2.5eV Photons3.0eV Photons

FWHM = 0.69eV

0.47eV

WHM = 0.41eV

emission, (2) 2.0 eV (622 nm) photons, (3) 2.25 eV (550 nm) photons, (4) 2.5 eV

e able to produce more current and at the same time reduce the energy spread from

engths the energy distribution spreads quickly.

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graph represents what is achievable with thermal-field emission,and the bottom right is what we hope to achieve with PHAST. Themost striking feature is the increase in brightness with either adrop or zero increase in energy spread; this is simply due to theenergy location of the photoelectrons. Each mark on the photoassisted solid lines corresponds to an increase in photon energy of0.2 eV. Initially there is very little increase in the brightness and sothe points are close together. Eventually we get significanttunneling and then emission over the barrier; once the photonenergy is roughly 0.4 eV above the Schottky barrier the energyspread races away. This is nicely demonstrated in Fig. 12.

4. Discussion

We can see from our simulations that the model we are usingis reasonable; it fits almost perfectly with the experimental dataof cesiated tungsten in Fig. 10 and follows roughly the curve fordifferent metals shown in Figs. 8 and 9. Fowler’s approximation ofadding the photon energy only to the normal component of theelectron energy is clearly reasonable. Dubridge also used this todescribe the energy distribution and showed that this nicelyreproduces experimental data [6]. From the work of Jensen wemay now include electron tunneling and obtain quantitativeresults. In this paper we have arrived at an expression (Eq. (24)),based on their work for the reduced on axis (maximum)brightness of a photofield emitter. This expression does not sufferfrom Fowler’s approximation and is in this sense exact.

However, the increase in brightness and control of energyspread in Fig. 11 which are so promising are only available atextremely high laser intensities around 10 TW=m2. With a CWlaser we cannot achieve the high laser intensities needed becausewe quickly calculate that the tip will be well above 1800 K (themanufacturers recommended operating temperature). In fact wecan roughly calculate that if we laser heat a Schottky tip to 1800 Kwe achieve only a photo-induced reduced on axis brightness B00ph

of 4� 104A=m2 sr V, about 104 times less than the thermal fieldreduced on axis brightness B00tf .

We therefore conclude that a radical design change is required.

4.1. Design change

It has long been known that a side illuminated tip should havea better quantum efficiency than a front illuminated mainly fromthe polarization sensitivity of the electronic surface states.Experimental evidence suggests that an increase of up to 7 timesmay be possible [27]. However, this is still not enough to get usthe factor 104 needed to compete with the standard Schottkysource.

We anticipate that by applying a thin film of tungsten to adiamond substrate and super cooling the diamond we may be ableto increase the power applied by at least 2 orders of magnitudewithout significant problems.

Further solutions lie in the use of materials with higherthermal conductivities or inherently better quantum efficienciessuch as semi-conducting materials.

For the present time we think that the primary application ofPHAST is in the pulsed regime with pulse times sufficiently shortto prevent overheating of the tip.

5. Conclusions

PHAST presents a new and exciting idea for an electron sourcewith higher brightness and lower energy spread than state of theart thermal-field emitters. Also the use of photons with energies

less than the barrier height (photofield emission) can make auseful contribution to the brightness via tunneling. Addingphotoemitted electrons does not necessarily damage the energyspread and may sometimes even improve it, merely by thelocation on the energy scale of the emitted photoelectrons. Thisopens up the use of both green and red lasers which areconsiderably cheaper than their UV cousins.

We hope that with more radical design and engineering wemay overcome the heating problems. However, it is uncertain asto how non-linear effects may cause problems.

We have shown that we are using a valid theory of electronemission which follows well the experimental results, and allowsus to make useful predictions.

To continue further theoretically, we hope to move to a fullyquantum description of emission and thus include polarizationsensitivity and band structure. Experimentally we will firstexplore the pulsed mode, allowing us to verify the theoreticalpredictions, which will also have applications in for example ultrafast microscopy.

Acknowledgment

Thanks to N.A. Moody for supplying the data contained in Fig.10.

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