Download Chapter 2 Enhancements

Ch.2 Enhancements
E2.1
Thermionic Electron Emission
Thermionic emission occurs when enough heat is supplied to the emitter that electrons
can overcome the work function energy barrier Ew of the material and escape into the vacuum.
Figure E2.1 illustrates the concept of overcoming the work function for the situation when no
strong extraction electric field is present in the gun. The symbol E represents the energy (work)
necessary to place an electron in the vacuum from the lowest energy state in the metal. But
electrons in a metal have a range of energies, and the highest energy state is called the Fermi
level, EF. When the emitter material is heated to a high temperature, a fraction of the electrons at
the Fermi level acquire enough energy to overcome Ew and escape into the vacuum. The cathode
current density Jc obtained from an emitter by thermionic emission is expressed by the
Richardson equation:
Jc  AcT exp(Ew / kT)
2
(E2.1)
where Ac = 120 A/cm2K2 is a constant for thermionic emitters (Heinrich, 1981), T (K) is the
absolute emission temperature, Ew (eV) is the work function of the filament material, and k is
Boltzmann's constant (8.6x10-5 eV/K). Both the temperature T and the work function Ew have a
strong effect on the emission current density Jc obtainable from the filament. Because it is
desirable to operate the electron gun at the lowest possible temperature to reduce evaporation of
the filament, materials of low work function are desired. Tungsten wire is often the material of
choice because it has a relatively low Ew and it produces a high value of Jc at temperatures well
below its melting point. An example calculation using Equation E2.1 may be obtained by
substituting typical values for tungsten: Ac = 120 A/cm2K2 , T = 2700 K, and Ew = 4.5 eV. The
resulting cathode current density is Jc = 3.4 A/cm2.
The lanthanum hexiboride (LaB6) filament also operates as a thermionic electron source.
From Equation E2.1 we note that the cathode current density Jc can be increased by lowering the
work function Ew. Because Ew is in the exponent, the effect can be dramatic. For example, at a
temperature of 2700 K, each 0.1 eV reduction in Ew increases Jc by about 1.5 times. Of the many
oxides and borides that exhibit low values of Ew, the most popular has been LaB6. The
measured work function of <100> single crystal LaB6 is about 2.5 eV (Swanson et al., 1981),
compared with 4.5 eV for tungsten. In fact, the current density produced by a tungsten filament
at 2700 K (3 A/cm2) may be obtained with LaB6 at 1600 K. But the typical operating
temperature for LaB6 filaments is 1800 K, for which the filament produces nearly 40 A/cm2. The
low operating temperature for LaB6 filaments increases emitter lifetime by reducing the material
evaporation rate, even while delivering a greater current density. In addition, a small radius (110 µm) at the cathode tip can enhance the effective local electric field at the tip (the Schottky
effect). This higher field can also lower the work function Ew confronting an electron escaping
the cathode. The amount by which the Schottky effect can reduce Ew is typically 0.1 eV or more
(Broers, 1975), leading to a further doubling of the cathode current density.
E2.2
More on Brightness
The Brightness Equation. The parameter of electron optical brightness incorporates the
current density but also recognizes changes in the angular spread of the electrons as they are
focused to form a beam crossovers at various points in the column. Brightness is defined as the
current density per solid angle and is given in an important relationship known as the brightness
equation:

4i b
 d 2 2
2
(E2.2)
where ib is the beam current at some point in the electron column outside the gun, d is the
diameter of the beam at this point, and  is the beam convergence (or divergence) angle at this
point. The dimensionless unit of solid angle is the steradian (sr) which may be thought of as an
ice-cream-cone-like section cut from a sphere. The solid angle in steradians may be calculated as
2 where  is the angle from the cone surface to its centerline.
Neglecting lens aberrations,
electron beam brightness is a constant throughout the electron column as the individual values of
ib, d, and  change. However, measurements of brightness made at the specimen level can only
be estimates of the actual gun brightness since the action of lens aberrations tends to lower the
brightness value measured at the specimen. Since brightness is the most important performance
indicator for electron guns, even relative estimates are valuable.
Maximum theoretical brightness. For thermionic emitters at high voltages, the maximum
theoretical brightness according to Langmuir (1937) is
max 
J ceV o
kT
(E2.3)
where Jc is the current density at the cathode surface (given by Equation E2.1), V0 is the
accelerating voltage (in volts), e is the electronic charge (1.59x10-19 C, coulomb), k is
Boltzmann's constant (8.6x10-5 eV/K), and T is the absolute temperature (K). An important
result from Equation E2.3 is that brightness increases linearly with accelerating voltage.
Substituting typical values for a tungsten gun operated at 2700 K and 20 kV, the maximum
theoretical brightness is 9.2 x104 A/cm2sr. Uncertainties in Jc from Equation E2.1 can easily lead
to errors in max of a factor of two.
Effect of bias voltage. To achieve the highest brightness for a given triode electron gun,
the bias voltage between the filament and the grid cap must be optimized by adjusting the bias
resistor. Emitted electrons will only proceed toward positive electric fields. At low bias, the
electrons enter a positive field over a large area of the filament tip, providing a a larger emission
current but a poor focusing action (Figure E2.2a). For this case, the emission current ie is high,
as shown in Figure E2.3 for a bias voltage of -300 V. Because of the weak focusing at low bias,
the crossover inside the gun d0 is large and the brightness  is not optimum. At very high bias,
only a negative field potential exists in front of the filament so electrons emitted are forced to
return to the filament (Figure E2.2c). Under high bias conditions (cut-off) both the emission
current and the brightness decrease to zero. Somewhere in between these two extremes lies an
optimum bias condition (Figure E2.2b) which will provide good emission, good focusing, and a
maximum in brightness (at about -400 V in Figure E2.3). Haine and Cosslett (1961) showed that
this optimum brightness is close to the maximum theoretical brightness for T = 2700 K. The
optimum brightness condition can also be altered by changing the filament-to-grid cap distance.
Usually the filament-to-grid cap spacing and the bias voltage are both fixed by the SEM
manufacturer. These preset values presumably correspond to the maximum brightness attainable
for that particular electron gun.
E2.3
More on Field Emission
Electron emission from a sharp tip. The field emission cathode is usually a sharp tip
fashioned from a single crystal tungsten wire that has been spot welded to a tungsten hairpin.
The significance of the small tip radius, about 100 nm or less, is that an electric field can be
concentrated to an extreme level at the tip. If the tip is held at a negative 3-5 kV relative to the
first anode, the effective electric field F at the tip is so strong (> 107 V/cm) that the potential
barrier for electrons becomes narrow in width as well as reduced in height by the Schottky effect.
This narrow barrier allows electrons to "tunnel" directly through the barrier and leave the cathode
without requiring any thermal energy to lift them over the work function barrier (Gomer, 1961).
Tungsten is the cathode material of choice since only very strong materials can withstand the
high mechanical stress placed on the tip in such a high electrical field. A cathode current density
as high as 105 A/cm2 may be obtained from a field emitter compared with about 3 A/cm2 from a
tungsten hairpin filament. In a field emitter, electrons emanate from a very small virtual source
(~10 nm) behind the sharp tip into a large semi-angle (nearly 20˚ or about 0.3 rad) which still
gives a high current per solid angle and thus a high brightness. A second anode is used to
accelerate the electrons to the operating voltage.
Brightness and probe current. The cathode current density Jc for a cold field emission
source strongly depends on the applied field strength. Expressions derived to calculate Jc
(Fowler and Nordheim, 1928; Good and Muller, 1956) yield values in the range Jc = 104 - 105
A/cm2. The maximum theoretical brightness of a field emission gun is given (Troyon, 1984) by
max 
J ceV 0
E
(E2.4)
where E, the energy spread of the beam, is only about 0.3 eV for cold field emission (Crewe et
al, 1971). Substituting E = 0.3 eV, Jc = 105 A/cm2 , and 20 kV into Equation E2.4, yields max
~ 2 x 109 A/cm2sr which is 102-103 times greater than the maximum brightness possible from a
thermionic source. Working back through the brightness equation (Equation E2.2), ignoring lens
aberrations, and assuming typical values for the beam parameters at the specimen (dp = 2 nm and
p = 7x10-3 rad), leads to an estimated maximum probe current for a cold field emission gun of
(ip)max ~ 10 nA. However, the measured brightness for cold field emission sources is typically
only about 108 A/cm2sr at 20 kV, so practical values of (ip)max are typically less than a few nA.
This maximum probe current is orders of magnitude less than the current possible from a
thermionic electron gun, but it is contained in a beam spot only 1-2 nm in diameter.
In order to increase the current that actually gets through the first anode aperture, several
methods have been developed to direct more current into a smaller angle. For a cold field
emitter, the orientation of the tungsten single crystal wire is usually <310> so that the direction of
minimum work function (Ew = 4.35 eV) is directed toward the anode (Orloff, 1984). Several
other methods have been used to confine the angle of emission from the tip: reshaping the tip by
heating while applying a field (Crewe et al., 1968; Swanson and Crouser, 1969), adsorption of
oxygen to induce reshaping by evaporating WO3 (Veneklasen and Siegel, 1972), and adsorption
of ZrO on the tip to reduce the work function (Swanson, 1975). In addition, electrostatic (Butler,
1966) and magnetic (Kuo, 1976) focusing elements have been placed between the first and
second anode to increase the current through the second anode aperture. The most successful
methods of increasing the total probe current from field emission guns have involved focusing
elements in the gun combined with the adsorption of ZrO on the <100> facet of a <100>-oriented
W tip (Tuggle et al., 1985). The ZrO lowers the work function Ew from 4.5 eV to 2.8 eV, and the
relatively large, flat emitting area of the <100> facet provides good emission stability. Figure
E2.4 shows an energy level diagram for three types of field emitters. Note that the Schottky
emitter (SE) still needs to be thermally assisted for electrons to overcome the energy barrier. The
thermal field emitter (TF) shown was an early configuration that is rarely used today. The cold
field emitter (FE) is represented by a curve that is so narrow that electrons can tunnel through the
barrier into the vacuum.
E2.4
More on Magnetic Lenses
A simple iron electromagnet producing a magnetic field is shown in Figure E2.5 where
the field across the gap is created by a current I (usually a few amps) energizing a coil of N turns.
This device produces a homogeneous magnetic flux (magnetic field) at the center of the gap. An
electron passing through this field perpendicular to the field lines moves in a curved trajectory
out of the paper because there is a force on the electron created by the electron moving in the
magnetic field. At the edge of the gap, the flux lines are bent. These flux lines are called the
"fringing field,” and it is this fringing field that is important in focusing electrons in a magnetic
electron lens.
Magnetic fields in the lens gap. An electron lens is a rotationally symmetric
electromagnet with the coil windings on the inside of the iron casing as shown in Figure E2.6.
Figure E2.7a shows the fringing magnetic field crossing the gap inside a schematic cross-section
of a magnetic lens. The magnetic flux density B denotes the intensity of the field. The symbol H
is often used for magnetic field strength, but in vacuum H=B so we will use only the symbol B.
Along each flux line, B is represented by a vector having both magnitude and direction. To
understand how moving electrons interact with the field B, it is useful to separate B into its
component vectors Br in the radial direction and Bz along the optic axis. Figure E2.7b shows that
Br reverses in direction as the flux line curves back to the lens, whereas Bz goes through a
maximum at the center of the lens gap.
Electron focusing. The equation that relates the force on the electron F to the velocity of
the electron v and the magnetic flux density B is
F  e(v  B)
(E2.5)
where e is the charge on the electron and the multiplication operation is the vector cross product
of v and B according to the "right hand rule" as shown in Figure E2.8. The “right hand” rule is
typically stated for a positive electric charge, but in this case the minus sign indicates that the
force on an electron is opposite that for a positive charge. Figure E2.8 shows that the force on a
positive charge is along the thumb when the index finger represents the particle velocity v and the
second finger represents the magnetic flux density B. The actual force on the electron is in the
direction opposite that of the thumb because the electron is negative.
Electron focusing occurs because the electron interacts with Br and Bz separately as
shown in Figure E2.9. An electron of velocity vz enters the lens parallel to the optic axis and
interacts with the radial component Br. According to the right hand rule, the vector product - e
(vz x Br) produces a rotational force into the paper Fin which in turn gives the electron a
rotational velocity vin. This rotational velocity then interacts with the axial component of the
field Bz to produce a radial force on the electron Fr = -e (vin x Bz). It is this radial force that
causes the electron trajectory to curve toward the optic axis and cross it. The focal length f of the
lens is the distance along the optic axis from the point where an electron first begins to change
direction to the point where it crosses the axis. Note that the actual trajectory of the electron as it
traverses the lens will be a spiral. The final image shows this spiraling action as a rotation of the
image as the objective lens strength is changed.
Lens current changes focal length. We can control the lens by changing the current I in
the lens coil which changes the focal length f. For most lenses the focal length f is nearly
proportional to V0/(NI)2 where N is the number of turns in the lens coil and V0 is the accelerating
voltage (Liebmann, 1955). Thus, the focal length f decreases as the current I increases, making
the lens stronger. Note also that the focal length will become longer at higher accelerating
voltages (higher kV) for the same lens excitation, since the velocity of the electrons increases
with increasing beam voltage. All modern SEMs automatically adjust I as a function of
accelerating voltage to compensate for this effect.
E2.5
Calculation of Electron Probe Diameter and Electron Probe Current
The general approach to controlling the aberrations in SEM lenses is to determine the
objective lens aperture angle opt, which minimizes the effect of the aberrations on the final
probe size. Use of opt also provides the maximum amount of current in the minimum probe
size.
Calculation of dmin and imax. The most important figure-of-merit for an SEM is the
amount of current that can be placed in small electron probes used for imaging and analysis.
Following Smith (1956), it is possible to calculate the diameter dp of the electron probe
impinging on the sample that carries a given current ip. Calculations of dp are made for the fullwidth-at-half-maximum (FWHM) of a gaussian intensity distribution which contains about half
of the total probe current. To obtain dp (FWHM), it is assumed that all significant aberrations
are caused by the final (objective) lens. To calculate the final probe size, the diameter of the
gaussian probe dg and the various aberration disks are assumed to be error functions so they can
be added together to produce dp as the square root of the sum of the squares of the separate
diameters,
d p  (dg2  ds2  dd2  dc2 )1/ 2
(E2.6)
where dg is the gaussian probe size at the specimen, ds is the spherical aberration disk, dd is the
aperture diffraction disk, and dc is the chromatic aberration disk. These quantities are given by
 4i 
dg   2p 2
  
(E2.7)
1
ds  Cs 3
2
(E2.8)
1/ 2
dd 
dc 
0.61
(E2.9)

E
C
E0 c
(E2.10)
Substituting these equations into Equation E2.6, we obtain
1/ 2
2
 4i p

1 2 6 (0.61)2 
E

2
d p   2 2  ( Cs)  
  Cc   
2
2

E 0  
 
(E2.11)
Lens current instabilities contributing to the chromatic aberration term are assumed to be
negligible in Equation E2.11. To find the optimum aperture angle that produces the smallest
electron probe, we first rearrange the brightness equation to solve for current:
ip 
 2 2 dg
4
(E2.12)
Approximation for normal accelerating voltages. For typical 10 to 30 kV operation, we
can neglect the chromatic aberration term and consider the case of a probe limited by spherical
aberration and diffraction effects only. The appropriate expression for the gaussian probe
diameter to substitute into Equation E2.12 can be obtained from
dg2  d 2p  ds2  dd2
(E2.13)
Next we substitute Equations E2.8, E2.9, and E2.13 into Equation E2.12. By differentiating the
resulting equation with respect to , we obtain an expression for dI/d. Solving for  in the
differentiated equation when dI/d = zero gives the result:
d 
opt   p 
Cs 
1/ 3
(E2.14)
Substituting Equation E2.14 into both Equation E2.11 (neglecting chromatic aberration for the
range 10 to 30 kV) and Equation E2.12 yields the following results:
 i

dmin  KC   p 2  1


3/ 8
1/ 4
s
imax
3/ 4
8/ 3
3 2 d p

 2/ 3
16 Cs
(E2.15)
(E2.16)
where K is a constant close to unity. The value of dmin is given in nanometers when Cs and  are
in nanometers, and ip is given in amperes (A). Equation E2.15 shows that the minimum probe
size dmin, decreases as the brightness  increases and as the electron wavelength  and spherical
aberration coefficient Cs decrease. In fact, in the limit of zero probe current, dmin reaches a value
of Cs1/43/4, which can be regarded as a measure of the theoretical resolution of the microscope.
From Equation E2.16, we can see that the maximum probe current varies as the 8/3 power of the
probe diameter. Since backscattered electron emission, secondary electron emission, and x-ray
emission all vary directly with probe current, these signals will fall off very rapidly as the probe
diameter is reduced (thermionic electron gun). More detailed expressions equivalent to
Equations E2.15 and E2.16 have been published (Pease and Nixon, 1965; Wells, 1974).
E2.6
Measuring Electron Beam Parameters
All of the parameters characterizing the electron beam incident on the sample (i.e., the
incident probe current ip, the probe diameter dp, and the convergence angle p) can be
experimentally determined. While there is no need to monitor such quantities continuously, it is
valuable to associate particular values of ip, dp, and p with specific operating conditions, both as
a means of setting these parameters for various operating modes and as a diagnostic tool in the
event of problems with the microscope.
Probe Current. The most straightforward quantity to measure is the incident beam current
onto the specimen ip since this can be done with a "Faraday cup" which is simply a container
completely closed except for a small entrance aperture (see Figure E2.10). An electron
microscope aperture disk of Pt (3 mm diameter) with a hole 25-100 µm in diameter is convenient
for this purpose. The container should be made from a material (e.g., Ti) different from that used
to fabricate the microscope stage. In this way any x-rays produced by stray electrons falling
outside the focused beam can be easily detected. The Faraday cup does not allow the
backscattered and secondary electrons generated by the incident beam to escape. The current
flowing to ground from the cup is therefore exactly equal to the incident beam current ip, and it
can conveniently be measured with a dc picoammeter, a calibrated specimen current amplifier, or
even a sensitive digital multimeter. For cases where the highest accuracy is not required, a flat
carbon block may be substituted for the Faraday cup. In this case the measured specimen current
isc and incident beam current ip are related as ip = isc/[1-( + )] where  and  are the backscatter
and secondary electron yields (Chapter 3), respectively. For a carbon sample normal to the beam,
both  and  are small enough that the measured specimen current is about 90% of ip.
Probe Size. The probe diameter may be measured by sweeping the beam across a sharp,
electron-opaque edge and observing the change in signal as a function of beam position. The
profile has the form shown in Figure E2.11. This image can usually be viewed or recorded from
the waveform mode of the scan generator. Typically the beam diameter is taken as the horizontal
distance between the 10% and 90% signal levels. Suitable sharp edges are clean razor blades,
cleavage edges in materials such as silicon, and fine-drawn wires such as fuse wire. The edge
must be clean, smooth, and non-transmitting to electrons. While these conditions are easy to
satisfy for large probes (~ 1 µm diameter), it is difficult to fabricate suitable edges for smaller
probe diameters. Thus, this measurement is not recommended for electron probes smaller than
about 0.1 µm. Ideally, the edge scanned should be over the entrance to a Faraday cup so that
none of the scatter of the incident beam modifies the profile.
Probe convergence angle. The probe convergence angle p can be obtained by measuring
the beam cross-sectional diameter at two points along the optic axis. The beam is focused on the
sharp edge and the "initial" probe diameter di is measured. Without changing the focus, the test
edge is moved vertically downward a known z-distance, say 1mm, using the calibrated z-control
of the SEM stage. At this position the larger "final" diameter df is measured (see Figure E2.12).
Since di is often too small to reliably measure, it may be neglected. Thus, the convergence angle
 may be found from
p 
d f  di
2z

df
(E2.17)
2z
Equation E2.17 will work whether the aperture is a real objective lens aperture or a virtual
objective aperture. For the case of the real objective aperture located in the gap of a pinhole lens,
the convergence may also be obtained using the diameter DA of the aperture hole in the final lens
and the distance from this aperture to the specimen, approximately the working distance W (see
Figure E2.13). Thus,
p 
DA
2W
(E2.18)
Estimation of beam brightness. By measuring ip, dp, and p under the same set of
conditions (same kV, C1 lens setting, final aperture size), the brightness may be calculated using
Equation E2.2. Since the brightness is constant throughout the electron column (neglecting
aberrations), this calculated value of the brightness is also an estimate of the gun brightness.
However, since measurement of probe size is difficult below 1 µm and since convergence angles
depend upon probe size measurements for systems with a virtual objective aperture, brightness
measurements under small probe conditions are only approximate. Still, the calculation of
brightness is instructive. For example, assuming a 100 µm real objective aperture, typical
tungsten gun values for beam current, beam size, and beam convergence are ip = 0.1 nA, dp = 30
nm, and p = 5 mrad. The brightness equation gives:

4 1010 A
 30  10 m 5  10 rad
2
9
2
3
2
 2  10 A /cm sr
5
2
While this value may be in error by a factor of two or so, it serves as a useful check that the SEM
is attaining an appropriate level of performance.
References for Ch. 2 Enhancements
Broers, A. N. (1975). SEM/1975, 662.
Butler, T. W. (1966). In Proc. 6th Int. Cong. Elect. Micros., Kyoto 1, 193.
Crewe, A. V., D. N. Eggenberger, J. Wall and L. M. Welter (1968). Rev. Sci. Instrum. 39, 576.
Crewe, A. V., M. Isaacson and D. Johnson (1971). Rev. Sci. Instrum. 42, 411.
Fowler, R. H., and L. W. Nordheim (1928). Proc. Roy. Soc. London, A111, 173.
Gomer, R. (1961). Field Emission and Field Ionization (Harvard Univ. Press, Cambridge, MA).
Good, R. H., Jr., and E. W. Muller (1956). In Handbuch der Physik (S. Flugge, ed.) (SpringerVerlag, Berlin), p. 176.
Haine, M. E., and V. E. Cosslett (1961). The Electron Microscope (Spon, London).
Heinrich, K. F. J. (1981). Electron Probe Microanalysis (Van Nostrand, New York).
Kuo, H. P. (1976). Ph.D. Dissertation, Cornell University.
Langmuir, D. B. (1937). Proc. IRE 25, 977.
Liebmann, G. (1955). Proc. Phys. Soc. B 68, 737.
Orloff, J. (1984). SEM/1984, 4, 1585.
Pease, R. F. W., and W. C. Nixon (1965). J. Phys. E 42, 281.
Smith, K. C. A. (1956). Ph.D. Dissertation, University of Cambridge.
Swanson, L. W. (1975). J. Vac. Sci. Technol. 12, 1228.
Swanson, L. W. and L. C. Crouser (1969). J. Appl. Phys. 40, 4741.
Swanson, L. W. , M. A. Gesley, and P. R. Davis (1981). Surf. Sci. 107, 263.
Troyon, M. (1984). Proc 8th European Congress on Electron Microscopy, Budapest (A. Csandy,
P. Rohlich, and D. Szabo, eds.) 1, p. 11.
Tuggle, D. W., J. Z. Li, and L. W. Swanson (1985). J. Microsc. 140, 293.
Veneklasen, L. H., and B. M. Siegel (1972). J. Appl. Phys. 43, 1600.
Wells, O. C. (1974). Scanning Electron Microscopy (McGraw-Hill, New York).