CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 117901
Formula for the Probability of Secondary Electrons Passing over the Surface
Barrier into a Vacuum *
XIE Ai-Gen(谢爱根)** , XIAO Shao-Rong(肖韶荣), ZHAN Yu(詹煜), ZHAO Hao-Feng(赵浩峰)
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology,
Nanjing 210044
(Received 5 April 2012)
Based on a simple classical model that primary electrons at high electron energy interact with the electrons of
lattice by the Coulomb force, we deduce the energy of secondary electrons. In addition, the number of secondary
electrons in the direction of velocity of primary electrons per unit path length, 𝑛, is obtained. According to
the energy band of the insulator, 𝑛, the definition of the probability 𝐵 of secondary electrons passing over the
surface barrier of insulator into the vacuum and the assumption that lattice scattering is ignored, we deduce the
expression of 𝐵 related to the width of the forbidden band (𝐸𝑔 ) and the electron affinity 𝜒. As a whole, the 𝐵
values calculated with the formula agree well with the experimental data. The calculated 𝐵 values lie between
zero and unity and are discussed theoretically. Finally, we conclude that the deduced formula and the theory
that explains the relationships among 𝐵, 𝜒 and 𝐸𝑔 are correct.
PACS: 79.20.Hx
DOI: 10.1088/0256-307X/29/11/117901
Secondary electron yield is an important and
widely studied topic,[1−8] and the probability of secondary electrons passing over the surface barrier of the
insulator into the vacuum, 𝐵, is an important research
topic of secondary electron yield. Up to now, 𝐵 has
been inaccessible to measurement. Alig and Bloom[9]
have deduced the formula of 𝐵 related to the maximum secondary electron yield 𝛿𝑚 , the primary energy
of 𝛿𝑚 , i.e. 𝐸𝑚 , and the average energy required to produce a secondary electron in an insulator, i.e. 𝜀; and
a simple theoretical relationship of 𝐵 to 𝜒/𝐸𝑔 was
developed to define an upper boundary for the 𝐵 calculated in Ref. [9], where 𝐸𝑔 and 𝜒 are the width of
the forbidden band and electron affinity, respectively.
In this Letter, based on a simple classical model
that primary electrons at a high electron energy interact with the electrons of the lattice by the Coulomb
force, the energy of secondary electrons is deduced.
Moreover, the number of secondary electrons in the direction of velocity of primary electrons per unit path
length (i.e. 𝑛) is obtained. According to the energy
band of the insulator, 𝑛, the definition of 𝐵 and the
assumption that lattice scattering is ignored, the formula for 𝐵 related to 𝐸𝑔 and 𝜒 is deduced, and the
theory that explains the relationships among 𝐵, 𝜒 and
𝐸𝑔 is developed and proven to be correct. The formula
for 𝐵 deduced here is universal for the estimation of
𝐵 under the condition that primary electrons at any
energy hit on insulators.
Primary electrons at high electron energy enter the
emitter perpendicularly and travel along a straight
line; the minimum distance between the electrons of
the lattice and the primary electrons is 𝑠, and the time
when primary electrons pass the minimum distance
between the electrons of lattice and primary electrons
is zero, as shown in Fig. 1, the distance 𝑟 between the
electrons of the lattice and the primary electrons can
be expressed as follows:[10]
𝑟 = (𝑠2 + 𝑣 2 𝑡2 )1/2 ,
(1)
where 𝑣 is the velocity of primary electrons, and 𝑡 is
the time measured from the instant of minimum distance.
Based on the process of movement of primary electrons, the momentum of an electron of a lattice is perpendicular to the velocity of the primary electron and
can be expressed as follows:[10]
∫︁∞
𝑒2 𝑠𝑑𝑡
2𝑒2
∆𝑝 =
=
,
(2)
𝑣𝑠
(𝑠2 + 𝑣 2 𝑡2 )3/2
−∞
where 𝑒 is electronic charge.
The energy of secondary electrons, 𝐸1 , can be written as[10]
𝑒4
1
(∆𝑝)2 =
,
(3)
𝐸1 =
2𝑚
𝐸𝑝 𝑠2
where 𝐸𝑝 = 𝑚𝑣 2 /2 is the primary energy, and 𝑚 is
the electron mass.
We draw a cylindrical shell in the direction of velocity of the primary electrons, the radius of the cylindrical shell is in the range [𝑠, 𝑠 + ∆𝑠], the number of
secondary electrons produced in the cylindrical shell
per unit path length can be written as
∆𝑛 = (2𝜋𝑠∆𝑠)𝑁 = 𝜋𝑁 ∆(𝑠2 ),
(4)
where 𝑁 is the number of secondary electrons per unit
volume.
Based on Eqs. (3) and (4), the number of secondary
electrons produced in the cylindrical shell per unit
path length can be written as
* Supported
∆𝑛 =
𝜋𝑁 𝑒4
1
𝜋𝑁 𝑒4 ∆𝐸1
∆( ) =
.
𝐸𝑝
𝐸1
𝐸𝑝 𝐸12
by the Natural Science Foundation of Jiangsu Provincial Universities under Grant No 10KJB180004.
author. Email: xagth@126.com
© 2012 Chinese Physical Society and IOP Publishing Ltd
** Corresponding
117901-1
(5)
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 117901
The number of secondary electrons produced in the
cylinder whose radius is 𝑠, per unit path length, can
be written as
∫︁
∑︁
𝜋𝑁 𝑒4
𝑑𝐸1
.
(6)
𝑛=
∆𝑛 =
𝐸𝑝
𝐸12
into the vacuum, produced in the cylinder whose radius is 𝑠, per unit path length, can be written as
When primary electrons at a high electron energy hit
on the insulator, most of the secondary electrons can
be excited from the valence band. The role of the lattice is minor.[11] We assume that lattice scattering is
ignored, therefore, according to the energy band of the
insulator, as shown in Fig. 2, the minimum (i.e. 𝐸min )
is 𝐸𝑔 .
The secondary electrons excited at distances larger
than the effective escape depth can not be emitted
into the vacuum,[14] and the probability of secondary
electrons with energy 𝐸 > 𝜒 reaching the surface can
be written as[15]
𝜋𝑁 𝑒4
∆𝑛 =
𝐸𝑝
(𝐸𝑔 +𝜒+50)
∫︁
(𝐸𝑔 +𝜒)
𝑑𝐸1
.
𝐸12
𝑓 (𝑥) = exp(−𝛼𝑥),
(9)
(10)
where 𝑥 is the distance from the surface of emitter, and
1/𝛼 is the effective escape depth of secondary electrons
with energy 𝐸 > 𝜒.
According to Eqs. (9) and (10), the number of secondary electrons which can reach the insulator surface
barrier, produced in the distance range of 0 ≤ 𝑥 ≤
1/𝛼, can be written as
Electron of lattice
r
S
νt
𝑛𝑝1 =
∑︁
Primary electron
Fig. 1. Distance between the electron of the lattice and
the primary electron.
𝑛𝑝2
∫︁1/𝛼
=
𝑛𝑝1 exp(−𝛼𝑥)𝑑𝑥
0
Vacuum level
(𝑒 − 1)𝜋𝑁 𝑒4
=
𝑒𝛼𝐸𝑝
χ
Bottom of conduction band
(𝐸𝑔 +𝜒+50)
∫︁
(𝐸𝑔 +𝜒)
𝑑𝐸1
,
𝐸12
(11)
𝐸1 can be written as
Eg
Top of valence band
𝐸1 = 𝐸 + 𝐸𝑔 ,
Fig. 2. The energy band of the insulator.
Classically,[12] the electrons in the conduction band
of the insulator with 𝐸 > 𝜒 can pass over the surface
barrier into a vacuum, where 𝐸 is measured from the
bottom of the conduction band. On the basis of the
energy band of the insulator, the conservation of energy and the above assumption, the minimum average
energy required to produce a secondary electron which
can pass over the insulator surface barrier into vacuum
can be written as
𝐸minipi = 𝜒 + 𝐸𝑔 .
(12)
where 𝐸1 is measured from the top of the valence
band.
The probability of a secondary electron with energy 𝐸 reaching the surface escapes from the solid is
𝑃 (𝐸), classically, 𝑃 (𝐸) = 0 for 𝐸 ≤ 𝜒. For 𝐸 > 𝜒,
𝑃 (𝐸) can be written as[12]
𝑃 (𝐸) = 1 − (𝜒/𝐸)1/2 .
Based on Eqs. (11)–(13),
electrons which can pass
barrier into the vacuum,
range of 0 ≤ 𝑥 ≤ 1/𝛼, can
(𝑒 − 1)𝜋𝑁 𝑒4
𝑛𝑝 =
𝑒𝛼𝐸𝑝
(7)
(13)
the number of secondary
over the insulator surface
produced in the distance
be written as
(𝜒+50)
∫︁
[1 − (𝜒/𝐸)1/2 ]
𝑑𝐸.
(𝐸𝑔 + 𝐸)2
(14)
𝜒
Secondary electrons are conventionally defined as having exit energies 𝐸𝑠 ≤ 50 eV while backscattered electrons have 𝐸𝑏 > 50 eV,[13] therefore, based on the energy band of the insulator, the conservation of energy,
the definition of secondary electrons and the above
assumption, the maximum 𝜀 can be written as
𝐸max = 𝜒 + 𝐸𝑔 + 50.
Based on Eqs. (6)–(8), the number of secondary electrons produced in the cylinder whose radius is 𝑠 per
unit path length can be written as
𝑛ti =
(8)
Based on Eq. (6)–(8), the number of secondary electrons which can pass over the insulator surface barrier
∑︁
𝜋𝑁 𝑒4
∆𝑛 =
𝐸𝑝
(𝐸𝑔 +𝑥+50)
∫︁
𝐸𝑔
𝑑𝐸1
.
𝐸12
(15)
The probability of secondary electrons with energy
𝐸 > 𝜒 reaching the surface can be written as Eq. (10),
117901-2
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 117901
that is to say, only a part of the secondary electrons
with energy 𝐸 > 𝜒, produced in the distance range
of 0 ≤ 𝑥 ≤ 1/𝛼, can reach the surface. The probability of secondary electrons with energy 0 ≤ 𝐸 ≤ 𝜒
reaching the surface can be written as[15]
𝑓 (𝑥) = exp(−𝛼2 𝑥),
(16)
where 1/𝛼2 is the effective escape depth of secondary
electrons with energy 0 ≤ 𝐸 ≤ 𝜒, the energy of secondary electrons with energy 0 ≤ 𝐸 ≤ 𝜒 is lower than
the energy of secondary electrons with energy 𝐸 > 𝜒,
so 1/𝛼2 is larger than 1/𝛼;[16] only a part of secondary
electrons with energy 0 ≤ 𝐸 ≤ 𝜒, produced in the distance range of 0 ≤ 𝑥 ≤ 1/𝛼2 , can reach the surface.
Thus we can think that the number of secondary electrons reaching the surface is approximately equal to
the number of secondary electrons produced in the
distance range of 0 ≤ 𝑥 ≤ 1/𝛼. Therefore, according
to Eq. (15), the number of secondary electrons reaching the surface can be written as
𝑛𝑡 =
𝑛ti
𝛼
𝜋𝑁 𝑒4
=
𝛼𝐸𝑝
(𝐸𝑔 +𝑥+50)
∫︁
𝐸𝑔
𝜋𝑁 𝑒4 (𝜒 + 50)
𝑑𝐸1
=
.
𝐸12
𝛼𝐸𝑝 𝐸𝑔 (𝐸𝑔 + 𝜒 + 50)
(17)
On the basis of Eqs. (14) and (17), the formula for
𝐵 can be written as
𝑛𝑝
0.63𝐸𝑔 (𝐸𝑔 +𝜒+50)
𝐵=
=
𝑛𝑡
(50+𝜒)
𝜒+50
∫︁
1 − (𝜒/𝐸)1/2
𝑑𝐸.
(𝐸𝑔 +𝐸)2
𝜒
(18)
Alig and Bloom[9] have given the formula for 𝐵, i.e.
𝐵 = 2.5𝛿𝑚 𝜀/𝐸𝑚 ,
(19)
where 𝜀 is written as
𝜀 = 2.8𝐸𝑔 .
(20)
Seen from Table 1, the 𝐵1 calculated with Eq. (18)
and the parameters in Table 1 agree well with the
𝐵2 calculated with Eqs. (19) and (20). However, for
the insulators whose band gap is very narrow, such
as InSb, Cs3 Bi, Ge and PbS, the 𝐵2 calculated with
Eqs. (19) and (20) and the parameters in Table 1 are
much fewer than the 𝐵1 calculated with Eq. (18) and
with the parameters in Table 1, the reason for is that
the 𝜀 calculated with Eq. (20) and the parameters in
Table 1 are much fewer than the experimental values
of the insulators whose band gap is very narrow, for
example, the 𝜀 of InSb calculated with Eq. (20) and
the parameters in Table 1 are equal to 0.56 eV and
are much fewer than the experimental data equal to
1.2 eV.[17] For another example, the 𝜀 of PbS calculated with Eq. (20) and with the parameters in Table
1 is equal to 1.12 eV and is much smaller than the
experimental values which are equal to 2.0 eV.[18]
As shown in Table 1, the 𝐵2 of MgO calculated
with Eqs. (19) and (20) and with the parameters in
Table 1 is much larger than the 𝐵1 of MgO calculated
with Eq. (18) and with the parameters in Table 1 it is
greater than unity. Because it is impossible for 𝐵 to be
larger than unity, we think that the 𝐵2 of MgO calculated with Eqs. (19) and (20) and with the parameters
in Table 1 is not reasonable and that the 𝐵1 of MgO
calculated with Eq. (18) and with the parameters in
Table 1 is reasonable.
Table 1. The calculated 𝐵1 and 𝐵2 .
Insulator 𝐸𝑔 (eV)[9] 𝜒 (eV)[9] 𝐵1 𝛿𝑚 [9] 𝐸𝑚 (eV)[9]
KF
10.9
0
0.63 11
1100
9.0
0.5
0.4 12
1700
NaCl
14
1200
11
1000
8.5
0.5
0.4
8
1500
KCl
13
1500
RbCl
8.3
0.5
0.4 11
1200
LiBr
8.0
0.2
0.47 8
1000
7.5
0.4
0.41 16
2200
NaBr
23
1600
7.6
0.8
0.34 10
1500
KBr
14
1500
13
1400
RbBr
7.2
0.4
0.41 11
1700
LiI
6.0
2.0
0.21 6
850
NaI
5.9
1.5
0.25 11
1100
11
1400
6.2
1.2
0.28 8
1600
KI
11
1500
RbI
5.8
1.2
0.27 10
1700
CsI
6.4
0.3
0.42 20
2600
7.8
0.9
0.33 24
1300
MgO
10
400
7.7
0.7
0.36 5
600
CaO
28
2600
BaO
5.1
0.6
0.34 4.7
450
ZnO
3.4
4.6
0.085 1.7
300
9.9
1.1
0.33 4
500
Al2 O3
6.4
650
SiO2
11
1.0
0.35 3
500
GaN
3.2
4.1
0.09 2.8
600
PbS
0.4
3.4
0.019 1.2
500
GaAs
1.4
4.0
0.049 1.1
350
InSb
0.2
4.6
0.007 1.15
700
C
5.5
6.3
0.087 2.8
750
Si
1.1
4.0
0.04 1.1
250
0.7
4.1
0.026 1.15
500
Ge
1.2
400
1.7
4.5
0.051 1.4
350
Se
1.4
400
Cs3 Sb
1.6
0.5
0.24 6
700
Rb3 Sb
1.0
1.2
0.11 7.1
450
Cs3 Bi
0.6
1.3
0.07 3.5
650
𝐵2
0.76
0.44
0.74
0.69
0.32
0.52
0.53
0.45
0.38
0.75
0.35
0.50
0.49
0.33
0.3
0.41
0.32
0.22
0.32
0.24
0.34
1.01
1.36
0.45
0.58
0.37
0.13
0.55
0.68
0.46
0.10
0.0067
0.031
0.0023
0.14
0.034
0.011
0.015
0.048
0.042
0.096
0.11
0.023
As seen from Table 1, the 𝐵1 values calculated
with Eq. (18) and with the parameters in Table 1 lie
between zero and unity and are consistent with predictions obtained using a free-particle approximation. It
is concluded that Eq. (18) and the theory that explains
117901-3
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 117901
the relationships among 𝐵, 𝜒 and 𝐸𝑔 are correct.
For a secondary electron emitter, 𝐵 is a constant
and is not a function of the incident energy of primary electrons.[1,2,4−6,15,19] Therefore, Eq. (18) for 𝐵
deduced under the condition that primary electrons at
a high electron energy hit on insulators is universal for
the estimation of 𝐵 under the condition that primary
electrons at any energy hit on insulators.
Based on a simple classical model that primary
electrons at high electron energy interact with the
electrons of lattice by the Coulomb force, the energy
band of the insulator, the conservation of energy and
the assumption that lattice scattering is ignored, we
successfully obtain Eq. (18) for 𝐵 and it is universal
for the estimation of 𝐵 under the condition that primary electrons at any energy hit on insulators. The
theory that explains the relationships among 𝐸𝑔 , 𝜒
and 𝐵 is correct.
References
[1] Xie A G, Zhang J and Wang T B 2011 Chin. Phys. Lett.
28 097901
[2] Xie A G, Li C Q and Wang T B 2009 Mod. Phys. Lett. B
23 2331
[3] Lei W, Zhang X B and Zhou X D 2005 Appl. Surf. Sci. 251
170
[4] Xie A G, Zhao H F, Song B and Pei Y J 2009 Nucl. Instrum.
Methods Phys. Res. Sect. B 267 1761
[5] Zhao S L and Bertrand P 2011 Chin. Phys. B 20 037901
[6] Xie A G, Zhang J and Wang T B 2011 Jpn. J. Appl. Phys.
50 126601
[7] Kazami Y, Junichiro K and Norio O 2009 Appl. Surf. Sci.
256 958
[8] Yasuda M, Tamura K and Kawata H 2001 Appl. Surf. Sci.
169 78
[9] Alig R C and Bloom S 1978 J. Appl. Phys. 49 3476
[10] Baroody E M 1950 Phys. Rev. 78 780
[11] Smith A C, Janak J F and Adler R B 1967 Electron. Conduction Solids (McGraw-Hill New York) P 235
[12] van der Ziel A 1968 Solid State Physical Electronics 2nd
edn (Englewood Cliffs: Prentice-Hall) p 172
[13] Reimer L and Drescher H 1977 J. Phys. D 10 805
[14] Copeland P L 1940 Phys. Rev. 58 604
[15] Seiler H 1983 J. Appl. Phys. 54 R1
[16] Nishimura K, Kawata J and Ohya K 2000 Nucl. Instrum.
Methods Phys. Res. Sect. B 164 903
[17] Tauc J and Abraham A 1959 Czech. J. Phys. 9 95
[18] Smith A and Dutton D 1958 J. Opt. Soc. Am. 48 1007
[19] Fijol J J, Then A M and Tasker G W 1991 Appl. Surf. Sci.
48 464
117901-4