Formulae for secondary electron yield from insulators and semiconductors

LOW ENERGY ACCELERATOR, RAY AND APPLICATIONS

Formulae for secondary electron yield from insulators and semiconductors

Ai-Gen Xie，

Min Lai，

Yu-Lin Chen，

Yu-Qing Xia

Nuclear Science and Techniques

Vol.28, No.10

Article number 141

Published in print 01 Oct 2017

Available online 06 Sep 2017

DOI：10.1007/s41365-017-0291-y

37600

The processes and characteristics of secondary electron emission in insulators and semiconductors were studied, and the formulae for the maximum yield (δ_m) at W_p0m≤800eV and the secondary electron yield from insulators and semiconductors δ at the primary incident energy of 2 keV≤W_p0<10 keV (δ_2-10) and 10 keV≤W_p0≤100 keV (δ_10-100) were deduced. The calculation results were compared with their corresponding experimental data. It is concluded that the deduced formulae can be used to calculate δ_2-100 at W_p0m≤800eV.

Maximum secondary electron yieldInsulators and semiconductorsSecondary electron yield.

1 Introduction

Secondary electron yields from different materials have interested many authors ^[1-4]. The secondary electron yield from insulators and semiconductors (δ) has found increasing applications in various areas, such as information technology, accelerator, scanning electron microscope, space flight etc. ^[5-7] However, due to the high resistance of insulators and semiconductors, it is difficult to overcome their surface charge problem and measure the secondary electron yield^[8-10]. Thus, available δ measurement data on insulators and semiconductors are not rich ^[11], especially in the incident energy range of 10–100 keV. Instead, many authors deduced formulae for key parameters to deduce δ of insulators and semiconductors^[12-16]; But up to now, no formulae is available for the incident energy range of 2–10 keV (δ_2-10) and 10–100 keV (δ_10-100), and for W_p0m, the incident energy at which δ is maximized (δ_m).

In this study, based on secondary electron emission processes in insulators and semiconductors and the formulae of δ_2-10, δ_10-100, and the maximum yield δ_m at W_p0m≤800 eV were deduced, involving the primary range of 10 keV≤W_p0≤100 keV (R_10-100) and 2 keV≤W_p0<10 keV (R_2-10), the atomic number Z, and the high energy back-scattering coefficient r, apart from the different forms of secondary electron yield. For compounds, Z represents average atomic number of compounds, for example, Z = (20+8)/2 for CaO; and r represents back-scattering coefficient in the energy range of W_p0≥10 keV.

2 Methods

2.1 Formula for δ_m

When electrons enter an insulator or semiconductor, secondary electrons are generated due to energy deposition of the primary electrons. Suppose that N(x,W_p) is the number of secondary electrons produced at a depth x and primary electron energy of W_p, N(x,W_p) is proportional to average loss of primary electrons per unit path length ^[17,18]:

N (x, W_{p}) = - \frac{d W_{p}}{d S} \frac{1}{ε},

(1)

where W_p is primary energy at given depth in insulator or semiconductor, S is the path length of primary electrons, and ε is the average energy required to produce an internal secondary electron in an insulator or semiconductor.

The probability of an internal secondary electron reaching the surface of the insulator or semiconductor and passing over the surface barrier into vacuum can be written as ^[17,18]:

f (x) = B exp (- α x),

(2)

where α is the absorption coefficient (and 1/α is mean escape depth of secondary electrons), and B is the probability that an internal secondary electron escapes into vacuum upon reaching the surface of insulator or semiconductor.

From Eqs. (1) and (2), the yield due to primary electron can be written as^[19]:

δ_{p} = \int_{0}^{R} N (x, W_{p}) B \exp (- α x) d x = - \frac{B}{ε} \int_{0}^{R} \frac{d W_{p}}{d S} \exp (- α x) d x,

(3)

According to Seiler ^[17], the maximum yield due to primary electron δ_pm is at R = 2.3/α. Then, δ_pm can be written as ^[19]:

δ_{pm} = - \frac{B}{ε} \int_{0}^{\frac{2.3}{α}} \frac{d W_{p}}{d S} \exp (- α x) d x .

(4)

At W_p0m≤800 eV, the primary range R can be expressed as ^[20]

R = 2 \times 10^{- 9} A_{α} W_{p0m} / (ρ Z^{2 / 3}),

(5)

where ρ is material density, Z is atomic number and A_α is atomic weight.

It can be assumed that the R at W_p0m≤800 eV approximately equals the corresponding S for deducing the formula for δ_m. Thus, the energy loss per unit path length of the primary electrons at W_p0m≤800 eV can be obtained by differentiating Eq.(5)

d W_{p} / d S = - ρ Z^{2 / 3} / (2 \times 10^{- 9} A_{α}) .

(6)

The δ_pm at W_p0m≤800 eV can be obtained by combining Eqs. (4) and (6)

δ_{pm} = (e^{2.3} - 1) B ρ Z^{2 / 3} / (2 \times 10^{- 9} A_{α} α e^{2.3} ε) .

(7)

Relation between δ_m and δ_pm can be written as ^[21]:

δ_{m} = (1 + 1.26 r) δ_{pm} .

(8)

For a given material, r is a constant ^[22]. The δ_m can be obtained by combining Eqs. (7) and (8).

δ_{m} = (1 + 1.26 r) (e^{2.3} - 1) B ρ Z^{2 / 3} / (2 \times 10^{- 9} A_{α} α e^{2.3} ε)

(9)

2.2 Formula for δ_10-100

When primary electrons of 10 keV≤W_p0≤100 keV enter an insulator or semiconductor, R_10-100 can be expressed as ^[20]

R_{10 - 100} = \frac{3.02 \times 10^{- 11} A α W_{p0}^{5 / 3}}{ρ Z^{8 / 9}} .

(10)

It can be assumed that the R_10-100 approximately equals the corresponding S for deducing the formula for δ_10-100. Thus, the following expression can be obtained by differentiating Eq.(10)

\frac{d W_{p}}{d S} = - \frac{ρ Z^{8 / 9}}{5.03 \times 10^{- 11} A_{α} W_{p}^{2 / 3}} .

(11)

The R_10-100 is much larger than the maximum escape depth of secondary electrons T. For example, using Eq.(10) in Si (Z=14, ρ = 2.35g/cm³,and A_α=28.1) ^[13] at W_p0=10 keV, we have R₁₀=16055Å and T = 41–54Å^[23]. Thus, most of the primary energy is dissipated outside T, and the primary energy changes little inside T. Then, the energy loss of primary electrons in the energy range of 10 keV≤W_p0≤100 keV per unit path length inside T can be approximately written as

\frac{d W_{p}}{d S} = - \frac{ρ Z^{8 / 9}}{5.03 \times 10^{- 11} A_{α} W_{p0}^{2 / 3}} .

(12)

The δ_p in energy range of 10 keV≤W_p0≤100 keV can be obtained by combining Eqs. (3) and (12)

δ_{p} = \frac{B}{ε} \frac{ρ Z^{8 / 9}}{5.03 \times 10^{- 11} A_{α} W_{p 0}^{2 / 3}} \int_{0}^{R} \exp (- α x) d x .

(13)

The R_10-100 is much larger than T. The internal secondary electrons excited outside T cannot be emitted into vacuum ^[23], and T approximately equals 5/α ^[23]. Thus, the definite integral [0,R] of Eq.(13) can be replaced with [0,5/α]. Then, we have:

δ = \frac{(e^{5} - 1)}{e^{5}} \frac{B}{α ε} \frac{ρ Z^{8 / 9}}{5.03 \times 10^{- 11} A_{α} W_{p 0}^{2 / 3}} .

(14)

The δ is composed of δ_p and yield due to backscattered electrons δ_r, i.e. ^[24]

δ = δ_{p} + η δ_{r} = (1 + β η) δ_{p} .

(15)

where β is the ratio of the mean secondary electron generation of one backscattered electron to that of one primary electron, and η is the back-scattering coefficient at W_p0. The β is greater than one^[17], because the larger average emission angle of backscattered electrons is more favorable for the excitation of secondary electrons than the normal incidence of primary electrons, and the mean energy of backscattered electrons is less than W_p0.

According to theoretical and experimental results, for 10 keV≤W_p0≤100 keV, β of metals is about 2 (β_10-100metal≈2) ^[25-28]. Because the average emission angle and mean energy of backscattered electrons from insulator or semiconductor are similar to those from metals^[28-29], we assume β_{10-100insulator} ≈2 for 10 keV≤W_p0≤100 keV. Therefore, we have:

δ_{10 - 100} = (1 + 2 r) δ_{p} .

(16)

The δ in energy range of 10 keV≤W_p0≤100 keV and W_p0m≤800 eV can be obtained by combining Eqs. (9), (14) and (16)

δ_{10 - 100} = \frac{4 3.9 (1 + 2 r) Z^{2 / 9} δ_{m}}{(1.1.26 r) W_{p0}^{2 / 3}} .

(17)

2.3 Formula for δ_2-10

When primary electrons of 2 keV≤W_p0<10 keV enter an insulator or semiconductor, R_2-10 can be expressed as ^[20]

R_{2 - 10} = \frac{1.03 \times 10^{- 10} A_{α} W_{p 0}^{3 / 2}}{ρ Z^{5 / 6}} .

(18)

It can be assumed that the R_2-10 approximately equals the corresponding S during for deducing the formula for δ_2-10. Thus, we have Eq.(19)

\frac{d W_{p}}{d S} = - \frac{ρ Z^{5 / 6}}{1.545 \times 10^{- 10} A_{α} W_{p}^{1 / 2}} .

(19)

The R_2-10 is much larger than T in energy range of W_p0m<800 eV. For example, for Si (Z =14, ρ= 2.35g/cm³, A_α=28.1) ^[13], at W_p0=2000 eV, calculated with Eq. (18) we have R₂=1222 Å and T =41–54Å^[23]. Thus, most of the primary energy is dissipated outside T, the primary energy only has a little change inside T. Then, the energy loss of primary electrons in the energy range of 2 keV≤W_p0<10 keV per unit path length inside T can be approximately written as:

\frac{d W_{p}}{d S} = - \frac{ρ Z^{5 / 6}}{1.545 \times 10^{- 10} A_{α} W_{p0}^{1 / 2}} .

(20)

Based on Eqs. (3) and (20), δ_p in energy range of 2 keV≤W_p0<10 keV can be written as

δ_{p} = \frac{B}{ε} \frac{ρ Z^{5 / 6}}{1.545 \times 10^{- 10} A_{α} W_{p0}^{1 / 2}} \int_{0}^{R} \exp (- α x) d x .

(21)

R_2-10 is much larger than T in energy range of W_p0m<800 eV, the internal secondary electrons excited outside T cannot be emitted into vacuum^[23], and T ≈5/α^[23]. Thus, the definite integral [0, R] of Eq.(21) can be replaced with [0, 5/α], i.e.,

δ_{p} = \frac{(e^{5} - 1)}{e^{5}} \frac{B}{α ε} \frac{ρ Z^{5 / 6}}{1.545 \times 10^{- 10} A_{α} W_{p0}^{1 / 2}} .

(22)

From Eqs. (8) and (15), we have δ_p/δ_r= 1/(1.26r) for W_p0m≤800 eV, and from Eqs. (15) and (16), we have δ_p/δ_r = 1/(2r) for 10 keV≤W_p0≤100 keV. There is the tendency that η increases with W_p0 for W_p0≤10 keV ^[17], so there is the tendency that the δ_p/δ_r decreases with W_p0 for W_p0≤10 keV. Thus, the δ_p/δ_r ratio for 2 keV≤W_p0<10 keV is in the range of [1/(1.26r),1/(2r)], and we assume δ_p/δ_r =1/(1.5r) for 2 keV≤W_p0<10 keV. Then, approximately, the δ for 2 keV≤W_p0<10 keV can be expressed as:

δ = (1 + 1.5 r) δ_{p} .

(23)

Combining Eqs. (22) and (23), we have:

δ_{p} = \frac{(e^{5} - 1)}{e^{5}} \frac{B}{α ε} \frac{ρ Z^{5 / 6} (1 + 1.5 r)}{1.545 \times 10^{- 10} A_{α} W_{p0}^{1 / 2}} .

(24)

From Eqs. (9) and (24), the δ for 2 keV≤W_p0<10 keV and W_p0m≤800 eV can be expressed as

δ_{2 - 10} = \frac{14.29 (1 + 1.5 r) Z^{1 / 6} δ_{m}}{(1 + 1.26 r) W_{p0}^{1 / 2}} .

(25)

3 Results and discussion

Several approximations were made in deducing Eqs. (17) and (25). For example, for 2 keV≤W_p0<10 keV, energy loss of primary electron per unit path length inside T is from approximation. The energy loss decreases with W_p^[4,30]. The R_2-10 is much larger than T for W_p0m<800 eV. Thus, most of the primary energy is dissipated outside T, the primary energy changes little inside T. Therefore, the energy loss of primary electron can be calculated Eq.(20)..

The δ_10-100 and δ_2-10 calculated with Eqs. (17) and (25), respectively, and the δ_m^[31-32], Z, W_po and r are given in Table. 1. It can be seen that the calculation results of δ_10-100 and δ_2-10 agree well with the experimental data ^[31-32]. So, Eq. (17) can be used to calculate δ_10-100, and the assumption that β_{10-100insulator} ≈ 2 is reasonable. Also, for 2 keV≤W_p0<10 keV, Eq. (25) can be used to calculate δ_2-10 and the assumption that the ratio of δ_p /δ_r ≈1/(1.5r) is reasonable.

Comparison of the calculated δ for insulators and semiconductors with the experimental data

Materials^[17,21]	δ_m	W_po(keV)	Calculated	Measured
Si (Z =14, r =0.22)	1.594^[31]	3.0^[31]	0.6723	0.769^[31]
		5.0^[31]	0.5207	0.49^[31]
		10^[31]	0.3056	0.3431^[31]
		20^[31]	0.1925	0.216^[31]
		30^[31]	0.1439	0.138^[31]
Te (Z =52, r =0.41)	1.63^[31]	3.0^[31]	0.875	0.891^[31]
		5.0^[31]	0.6777	0.706^[31]
		10^[31]	0.4451	0.376^[31]
		20^[31]	0.2804	0.319^[31]
		30^[31]	0.214	0.174^[31]
	0.819^[11]	2.0^[11]	0.5384	0.495^[11]
		2.4^[11]	0.4915	0.418^[11]
Ge (Z =32, r =0.35)	1.142^[33]	2.0^[33]	0.688	0.696^[33]
		2.5^[33]	0.6154	0.54^[33]
		3.0^[33]	0.5618	0.53^[33]
		3.5^[33]	0.52	0.49^[33]
		4.0^[33]	0.4865	0.45^[33]
		4.5^[33]	0.4587	0.42^[33]
		5.0^[33]	0.4352	0.42^[33]
	0.743^[11]	2.0^[11]	0.4477	0.398^[11]
		3.0^[11]	0.3655	0.312^[11]
		4.0^[11]	0.3165	0.264^[11]
	1.62^[31]	5.0^[31]	0.6174	0.649^[31]
		20.0^[31]	0.2458	0.228^[31]
C (Z =6, r =0.1)	1.56^[31]	5.0^[31]	0.434	0.501^[31]
		20^[31]	0.1475	0.125^[31]
	1.136^[33]	2.0^[34]	0.5	0.527^[34]
		2.5^[34]	0.447	0.442^[34]
		3.0^[34]	0.408	0.4^[34]
		4.0^[34]	0.3533	0.316^[34]
		5.0^[34]	0.316	0.315^[34]
Indium tin oxide (Z =35.7, r =0.36)	2.52^[32]	2.0^[32]	1.5477	1.55^[31]
		3.0^[32]	1.2637	1.31^[32]
		5.0^[32]	0.9789	1.03^[32]
		7.0^[32]	0.8273	0.72^[32]
		10^[32]	0.6356	0.73^[32]
Indium zinc oxide (Z =29, r =0.34)	2.67^[32]	2.0^[32]	1.581	1.61^[32]
		3.0^[32]	1.291	1.31^[32]
		5.0^[32]	1.0	1.02^[32]
		7.0^[32]	0.845	0.86^[32]
		10^[32]	0.6278	0.72^[32]
Al₂O₃ (Z =10, r =0.18)	6.23^[35]	2.0^[35]	3.025	3.83^[35]
		3.0^[35]	2.47	2.44^[35]
V₂O₅ (Z =12.3, r =0.2)	1.216^[36]	2.01^[36]	0.6114	0.718^[36]
		2.54^[36]	0.5439	0.669^[36]
		3.044^[36]	0.4968	0.578^[36]
		3.572^[36]	0.4586	0.53^[36]
		4.034^[36]	0.4316	0.487^[36]
		5.056^[36]	0.3855	0.389^[36]
		5.571^[36]	0.367	0.347^[36]

The secondary electrons inside an insulator or semiconductor lose their energy in electron-electron collisions. Classically, only electrons with a kinetic energy E>χ_real can escape into vacuum ^[37], where χ_real is the real electron affinity and E is the energy measured from the bottom of conduction band of insulator and semiconductor. The minimum energy (for secondary electrons to escape) increases with the χ_real. The electrons that are created in shallower depths suffer fewer collisions and survive with sufficient energy to escape. In other words, volume in the insulator or semiconductor from which electrons escape decreases with increasing χ_real, and from this reduced volume only hot electrons emerge. As a result, a decrease in 1/α is expected. With the increase of χ_real, the minimum energy for secondary electrons to escape increases, and B decreases. Therefore, from Eq. (9), δ_m increases with decreasing χ_real.

For an insulator or semiconductor, different sample preparation techniques can lead to different χ_real, hence different δ_m, and different δ_2-10 and δ_10-100, from Eqs. (17) and (25). These can be seen clearly for Te and Ge shown in Table 1 From above, it is concluded that the formulae for δ_10-100 and δ_2-10 can be used to calculate δ_2-100 in the energy range of W_p0m≤800 eV.

4 Conclusion

Based on the processes and characteristics of secondary electron emission, formulae for R_10-100 and R_2-10 and relationships among parameters of δ, the formulae for δ_10-100 and δ_2-10 were deduced, respectively. The δ_10-100 calculated with the deduced formula for δ_10-100 and δ_2-10 calculated with the deduced formula for δ_2-10 were compared with their corresponding experimental data, and the results were analyzed. It is concluded that the deduced formulae for δ_10-100 and δ_2-10 can be used to calculate δ_2-100 in the energy range of W_p0m≤800 eV.

Different sample preparation of an insulator or semiconductor can lead to different χ_real, 1/α, B, δ_m, δ_2-10 and δ_10-100.

References

[1]

J. Wang, Y. Wang, Y.H. Xu, et al.,

Research on the secondary electron yield of TiZrV-Pd thin film coatings

. Vacuum 131, 81-88(2016).doi: 10.1016/j.vacuum.2016.05.001

Formulae for secondary electron yield from insulators and semiconductors

1 Introduction

2 Methods

2.1 Formula for δm

2.2 Formula for δ10-100

2.3 Formula for δ2-10

3 Results and discussion

4 Conclusion

2.1 Formula for δ_m

2.2 Formula for δ_10-100

2.3 Formula for δ_2-10