PHYSICAL REVIEW B VOLUME 42, NUMBER 14 15NOVEMBER 1990-I
Theoretical
study
of
collimated
fieldemission
of
electrons
from
a
point
source
E.
Tekman andS.
CiraciDepartment
of
Physics, Bilkent University, Bilkent, 06533Ankara, Turkey A. BaratoffIBMResearch Division, Zurich Research laboratory, CH-8803 Ruschlikon, Switzerland (Received 26 June 1990;revised manuscript received 29August 1990)
We clarify basic mechanisms for collimated field emission of electrons from a metallic tip of atomic dimensions. The eA'ective potential barriers arising from the lateral confinement of current carrying states efficiently suppress states ~ith higher transverse quantum numbers. The
hornlike opening of the three-dimesional potential improves collimation even ifthe transmission is not adiabatic. We also find field-dependent resonance and diA'raction eA'ects.
Recently, Fink' achieved the fabrication
of
stable%(111)
tips terminated by three or even one atom, thus providing a charged-particle sources of atomic dimen-sions. A stable low-energy electron beam with currentI
up to—
10 pA can be obtained by using much lower volt-ages(a
few 100V)
than with conventional field emitters. Moreover, in spite of the finite transverse momentumof
the incident electrons, the beam is well collimated, its an-gular halfwidth being as small as-2'
at the collector screen. Very recently, interference fringes due to scatter-ingof
a similar beam by carbon fibers have demonstrated sufficient coherence to perform holography with low-energy electrons. 'A proper understanding
of
the eff'ects producing such a collimated and coherent electron beam has been the major issue. The collimationof
the electrons in semiconductor microstructures has been treated earlier in the contextof
point-contact resistances andof
Hall-efl'ect quenching. i In those systems, the dimensionality and, especially the (ballistic) character ofthe transport were, however, quite different from the present case. Recent theoretical stud-ies 'on field emission from atomic-size sources arrived at difl'erent conclusions for the mechanisms
of
collimation. In the studies by Garcia and co-workers the source was simulated by a quasi-one-dimensional constriction con-nected to a free-electron reservoir, and the effectof
the applied fieldF
was represented by a potential barrierof
constant height abutting this constriction. They found that the potential barrier isexclusively responsible for the collimation, a plane triangular barrier being particularly efficient. On the other hand, Lang, Yacoby, and Imrys noted that owing to the enhanced field near the tip, a po-tential channel with a hornlike profile forms in frontof
its apex. In this waveguide the single-particle wave functions were claimed to evolve adiabatically, so that their longi-tudinal momentum increases without reflections and scattering among subbands (modes). The effective barrierin the channel filters incident states by selecting those with small transverse momentum. Collimation was attri-buted to the barrier and to the adiabatic evolution
of
the states in the hornlike channel opening. By themselves, however, the self-consistent-field(SCF)
jellium calcula-tions by Lang et al. did not provide evidence foradiabati-city in situations where relevant dimensions are compara-ble to A,
F-3.
5 A. For a single-atom tip this occurs infields
F-1
V/A required for high currents.In this Rapid Communication, we present the results
of
a model study, which resolve those controversial issues and enable a systematic analysis
of
the influence of different parameters(e.
g.,F,
tip work function, channel width, and flaring) or relevant properties of the emitted beam. In particular, we clarify basic mechanisms leading to the collimation and find that the mode (subband) selec-tion by the effective barrier due totransverse confinement of current carrying states is indeed essential for atomic-size emitters. The electric field and the hornlike opening improve the collimation even though the adiabatic is not valid. In experiments, the electrode or screen towhich the voltage is applied is placed centimeters to fractionsof
pm away, but the intrinsic quantum phenomena of interest occur only within afew kF ofthe tip. Additional collima-tion due to the curvatureof
electron trajectories in the re-gion beyond isnot considered here.The three-dimensional
(3D)
potential in the vicinityof
the tip is represented byV(F;p,
z)
=tlt(F,
z)+a(z)p'8(z+l,
)8(d
—
z),
(1)
where p
(F,
z)
is the bimetallic junction potential calcu-lated for two parallel jellium electrodes with edges atz0
(tip) and z
L.
The electric field is incorporated follow-ing Orosz and Balazs. In the region between the baseof
the tip (z(
—
l,)
and the outer vacuum region where la-teral variations become negligible (z)
d),
the potential is assumed parabolic in the transverse direction, thus defining a channel in frontof
the apexof
the tip. As schematically illustrated in Fig. 1, Eq.(1)
can also de-scribe the hornlike shapeof
the potential on the tip side and on the vacuum side. This is achieved by uniformly varying the lateral extendof
the confining potential,w (l'i
/2ma)'t
in the intervals(
—
l„—
l~) and (lq,d).
Earlier, a similar typeof
potential forF
0
was derived fromSCF
pseudopotential calculations and used to ana-lyze the characterof
transport' as a functionof
tip-sample distance in scanning tunneling microscopy. The form ofthe potential Eq.(1)
is in compliance with the re-sultsof
SCF
calculations, as may be judged by compar-9221@1990
The American Physical Society9222
E.
TEKMAN,S.
CIRACI, AND A. BARATOFF Emitter T1P Ck', I -g, -g, o@.
(F, z) n=) ~~Z~eff~ ~~~e&&'n=0 E Screen y,(z)
{(2m/62)[E
—
y(F,z)
—
e„(z)]j»'
is the propagation constant for the nth (n
=n„+n~)
sub-band state quantized in the channel with energye„(z)
=(n+1)h
/mw(z)
and harmonic oscillator eigen-function4„(p,z)
a:exp[
—
p /2w(z)l.
The coefficients A„p and B„p, are determined by imposing the usual con-tinuity requirements at the boundaries between the seg-ments and at z—
I, and z=d.
The current energy dis-tribution is derived from the expectation valueof
the current-density operator,fO
J(F,E)
(2&) dk&(yk,.~j,
~yk,.)b(h
k;/2m—
E),
and its angular spread
T1P 1Ii I][ I
g
II]
~]
/'r
I//
I l / / / g / I ( J I IFIG. 1. Top: schematic illustration ofp
(F,
z),n-dependent Qff(F,z),
a(z),
and geometrical parameters. Bottom: contoursofconstant potential V(F;p,
z)
calculated in aplane p~
4.25 Aand
—
4&
z~
10A for a; 0.2, a, 0.5,ao 0.02 eV/Ai, andF
2V/k
The contour spacing is2.5eV; dashed lines are for V(F;p,z)
(EF.
Throughout this work geometrical parameters, I, 4 A, d 10 A, L 30 A, and electronic parameters,EF-12.
5 eV and tip work functionp-5
eV corresponding toaluminum jellium are used.
(2)
ing Fig. 5of Ref.
5 with the contoursof V(F;p,
z)
shown in Fig. 1 for particular parameter values. In realitya
in-creases and thus the lateral confinement becomes pro-nounced as the size
of
the tip apex isdecreased orF
is in-creased, but it is convenient toconceptually separate lon-gitudinal eff'ects contained in p(F,z)
from transverse ones. In the present model the channel modifies the po-tential barrier and the transmission in the presenceof
F
is treated quantum mechanically across and behind the bar-rier up to z d, and semiclassically beyond. By contrast, in the modelsof
Garcia and co-workers the channel pre-cedes the potential barrier in space, and the motion ofthe electrons is sometimes treated classically just beyond the turning point.Using the transfer-matrix method described by Tekrnan and Ciraci,
"
we obtain current carrying solutions qr&,corresponding to an incident wave vector k; deep in the emitter.
To
this end, we divide the region—
1,~
z~
d into discrete segments, in which p(F,z)
anda(z)
can be assumed constant. In each segment the wave functions have the form~(F
E),
„-
&'(E))
(2m/&
')
[E
p(F—
,z, )
l—
(x'(E))
(4)
is defined in termsof
the ex ectation valueof
the trans-verse wave vector squared(v
(E))
at apointz,
slightly to the leftof
zL.
Neglecting thermal broadening, the total emission current is expressed byI(F)
fo'dE
J(F,E);
the energy spreadof
the emitted beam is specified byfo'dE(EF
—E)J(F,
E)/1(F),
and its collimation angle by8,
(F)
-
f,
'dE
n
(F,
E)J(F,
E)/I(F)
.(5)
Note that collimation eff'ects due to eff'ective barrier, the horn, and the electric field are taken into account. On the other hand, the emission angle
8,
(F)
similarly defined at zd
excludes the semiclassical collimation effect due to the electric field beyond the horn.In order to reveal the effects
of
transverse confinement we first assume thata
in Eq.(1)
is constant throughout the range—
I,~
z~
d(i.
e., a uniform channel). In Fig.2(a)
the variation ofI
withF
is illustrated for several valuesof
a.
For fixeda,
I
increases with increasingF,
since the height and thicknessof
the tunneling barrier de-crease. At largeF,
p(F,
z)
iseventually depressed belowEF,
as illustrated in Fig. 1, the lowest effective barrier Pff(F,z)
—
[eo(z)+
p(F,
z)
—
EFl
almost disappears, sothat
I(F)
tends to level off. On the other hand, the effective emission area decreases and subbands (channel modes) shift upward in energy with increasinga;
conse-quently the corresponding effectivebarriers"
increase; this causesI
to decrease for fixedF.
The net effects
of
F
anda
on8,
are summarized in Fig.2(b):
the collimation is improved by decreasing confinement(i.
e., increasing the source size) and also by increasingF.
We attribute these trends to reduced diffraction atthe end ofthe channel and to the more rapid increaseof
the longitudinal wave vectory„p(z)
beyond the effective barrier. By themselves, the concomitant de-crease in barrier height and thickness would give rise to the opposite trends. Moreover, as emphasized inRef.
4, these trends would be unaffected ifthe channel preceded the barrier. Such a situaiton may well be realized insemi-THEORETICAL STUDYOFCOLLIMATED FIELD EMISSION
OF.
.
.
9223 2 10— 24— -4 10 -2 10— 1.2— O 0 10—8-
08-+=0005 4 1 t 2 F(V/A) 0 0.8 0.9 / 1.0FIG.2. (a) Emission current
I
and (b)collimation angle 8, at z z, vs electric fieldF
for uniform channels (a; a,ao
a).
Current energy distributionJ(F,E)
is shown for (c) a 0.005 eV/A', and (d) a 0.1eV/A'; the contribution offirst n 0(second n 1)mode is shown by the dashed (dotted) line.conductor microstructure, but does not apply in our case. It would be highly desirable totest these differing predic-tions. Care should be taken tomeasure a quantity similar to
8„e.
g., the angular width at half maximum. The ap-parent size ofthe beam spot above acertain detection lev-el may show spurious trends. One must also keep in mind that the above-mentioned increaseof
a
withF
might give rise to an increase in8„
followed by saturation at highF.
Furthermore, for a somewhat broader facet atthe apexof
the tip, the field will be enhanced at its edge or corners, so that our model no longer applies.The weak features in
8,
(F)
curves found at higha
andF
[shown by arrows in Fig.2(b)],
arise from matching to field resonances,i.e.
, approximately standing-wave solu-tion between the outer edgeof
the barrier and thepartial-ly reflecting channel end.
"'2
The resulting modulationof
the transmitted diffraction pattern gives rise to structure inQ(F,
E)
and8,
(F).
WhileJ(F,
E)
andI(F)
are de-voidof
any visible structure, d (log~oJ)/dE exhibits os-cillations shifting withF.
In Figs.
2(c)
and2(d)
the current energy distributionJ(F,
E)
is shown for widely different valuesof a,
together with the contributions from the lowest two subbands.It
is clear that the mode selection improves with increasinga
due tothe increased subband separation, (e~—
eo). In Fig.2(c)
then=1
contribution exceeds that from the n0
mode because the former is doubly degenerate. Since the energy spreadhE
is determined by tunneling, it increases with increasingF
and decreasinga (i.e.
, with increasingI).
A large current and a small AE (less than-0.
5 eV) are mutally exclusive. The effectof a
due to mode selec-tion within the barrier isessentially absentif
the channel precedes the latter. The effectsof
mode selection on col-limation are more subtle because8,
appears dominated by the growth of y,(z)
and bydiffraction beyond the barrier.We next consider the effects of the hornlike opening into the outer region, which is described by the
parame-2 10— 0 10— -2 10— 10 (a) 60-~I 40-C
0
o 20 (o) (e) n,=005 rn o'0=0015-—
1 1 (c) F(V/A)FIG.3.(a),(d) Emission current
I,
(b), (e)contributions from the n 0and n 1 modes evaluated atz=d.
(c),(f)collimationangle 8, at z z,for two horn geometries (see insets) shown by
the lines through crosses. The left- (right-) hand-side panel
cor-responds toa, 0.05 (a, 0.25);ao
=0.
01 eV/A',I2=4
A. Theopen (solid) circles correspond to the same quantities for the uniform channels with
a-ao
(a,).
ters
a„ao,
and 12in Fig.1.
These effects are examined by comparing the results obtained for two horn structures with those calculated for the corresponding uniform chan-nels witha
a, or ao, as shown in Fig.3.
The effective barrier with the horn is higher than that fora=ac,
but only slightly thinner than that fora
=a,
. ConsequentlyI
is significantly higher fora
ao, but approaches that ob-tained for the uniform channel witha
=a,
for largeF
as seen in Figs.3(a)
and3(d).
The contributionof
the two lowest modes (n0
and n1)
to the total emission current, calculated at z d, isillustrated in Figs.3(b)
and3(e).
Obviously, the suppressionof
the higher modes is more completeif
the channel is narrower either along its whole length or its central portion. The deteriorationof
mode selection with increasingF
arises because the ratio(y~
—
yo)/yo then becomes increasingly smaller beyond the higher effective barrier.Compared tothe uniform channel with
a
a„
the effectof
the horn on mode selection is small and more pro-nounced at lowF,
but the improvement in collimation ap-parent in Figs.3(c)
and3(f)
is quite dramatic. An analysisof
the relative contributions from different modes in different segments along the horn reveals that their ra-tio is nearly invariant. At first sight the explanationof
collimation by Lang et al. in termsof
mode selection in-side the barrier followed by nearly perfect transmission at9224
E.
TEKMAN,S.
CIRACI, AND A. BARATOFFits end appears correct. However, such an adiabatic evo-lution takes place ifthe variation
of
the channel with w is small on the scale ofthe electron wavelength. This is not the case, especially for the more flared horn with at-0.
25.If
the adiabatic picture were valid, the subband states (modes) would evolve slowly and independently. Hence, the total emission current and collimation angle could be obtained by summing only the diagonal matrix elements (with respect to the mode index n)of
the current-density operatorj,
in Eq.(3).
For a uniform channel the eigenfunctions4„(p)
are z independent and thus the diagonal approximation is exact (excluding effects due to refiections). On the other hand, because of the adiabatic reductionof
the transverse momentum x along the lengthof
aslowly varying horn, the diagonal ap-proximation isexpected to give8,
smaller than those cal-culated for the corresponding uniform channels witha
a,
anda
ao. In our example for the horn structure witha,
0.
25, the full calculation givesI
2.04
pA and8,
6.
2'
forF
1.
6V/A, whereas the diagonal approxi-mation yieldsI =1.
37 pA and8,
3.
6'.
The valuesof 8,
calculated for the uniform constrictions with
a=a,
anda=co
in Figs.3(c)-3(f)
are-8'
and 5.4',
respectively. This demonstrates that in the presenceof
the horn8,
is significantly increased owing to intersubband mixing. In viewof
this analysis andof
the results illustrated Figs.3(c)-3(f)
we argue that the hornlike potential profile typ-ical for an atomic-size, high-current source improves the collimation, although the adiabatic picture isnot valid.In conclusion, even ifthe adiabatic approximation does not apply, owing to mode selection near the apex
of
the tip and confinement in the channel extending beyond the bar-rier, our model reproduces observed propertiesof
beams emitted from atomic-size tips. We also found field-dependent resonance and diffraction effects.This work was supported by Joint Project Agreement between Bilkent University and the
IBM
Zurich Research Laboratory. We thank H. de Raedt,Y.
Imry,N.
Lang, andJ. J.
Saenz for supplying copiesof
their work prior to publication, and H.-W.
Fink,R.
Morin, andW.
Stocker for discussions.'H. W. Fink, IBM
J.
Res. Dev. 30, 460 (1986);Phys. Scr.38, 260 (1988);for the holography see H.W. Fink, W.Stocker,and H. Schmid, Phys. Rev.Lett. 65, 1204(1990).
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J.
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(1989).
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J. J.
Saenz, and H. de Raedt,J.
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Saenz, Phys. Rev. Lett. 63, 2260
(1989).
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(1989).
L.I.Glazman, G. B.Lesovik, D. E.Khmel'nitskii, and R.
I.
Shekter, Pisma Zh. Eksp.Teor. Fiz.48,218[Sov. Phys. JETP48, 238 (1988)l;A. Yacoby and Y.Imry, Phys. Rev. B41, 5341
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