Formulae for secondary electron yield from insulators and semiconductors

Formulae for secondary electron yield from insulators and semiconductors 1 corresp first-author Corresponding author E-mail:xagth@126.com Corresponding author E-mail:xagth@126.com E-mail:xagth@126.com 1 1 1 School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China 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 yield Insulators and semiconductors Secondary electron yield. 1 Introduction 1 Secondary electron yields from different materials have interested many authors [ 1 1 - 4 4 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 5 - 7 7 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 8 - 10 10 8-10 ] . Thus, available δ measurement data on insulators and semiconductors are not rich [ 11 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 12 - 16 16 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 1 2.1 Formula for δ m 2 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 17 , 18 18 ] : <math display="block" id="M1"><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mtext>p</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac><mfrac><mn>1</mn><mi>ε</mi></mfrac><mo>,</mo></mrow></math> 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 17 , 18 18 ] : <math display="block" id="M2"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>B</mi><mtext>exp</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mi>α</mi><mi>x</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> 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 19 ] : <math display="block" id="M3"><mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mi>R</mi></munderover><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mtext>p</mtext></msub><mo stretchy="false">)</mo></mrow></mrow></mstyle><mi>B</mi><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mi>B</mi><mi>ε</mi></mfrac><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mi>R</mi></munderover><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac></mrow></mrow></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>x</mi><mo>,</mo></mrow></math> According to Seiler [ 17 17 ] , the maximum yield due to primary electron δ pm is at R = 2.3/ α . Then, δ pm can be written as [ 19 19 ] : <math display="block" id="M4"><mrow><msub><mi>δ</mi><mrow><mtext>pm</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mi>B</mi><mi>ε</mi></mfrac><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mrow><mfrac><mrow><mn>2.3</mn></mrow><mi>α</mi></mfrac></mrow></munderover><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac></mrow></mrow></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>x</mi><mo>.</mo></mrow></math> At W p0m ≤800 eV, the primary range R can be expressed as [ 20 20 ] <math display="block" id="M5"><mrow><mi>R</mi><mo>=</mo><mtext> </mtext><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup><msub><mi>A</mi><mi>α</mi></msub><msub><mi>W</mi><mrow><mtext>p0m</mtext></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> 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) <math display="block" id="M6"><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub><mo>/</mo><mtext>d</mtext><mi>S</mi><mo>=</mo><mtext> </mtext><mo>−</mo><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup><msub><mi>A</mi><mi>α</mi></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> The δ pm at W p0m ≤800 eV can be obtained by combining Eqs. (4) and (6) <math display="block" id="M7"><mrow><msub><mi>δ</mi><mrow><mtext>pm</mtext></mrow></msub><mo>=</mo><mtext> </mtext><mrow><mo>(</mo><mrow><msup><mi>e</mi><mrow><mn>2.3</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>B</mi><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><mi>α</mi><msup><mi>e</mi><mrow><mn>2.3</mn></mrow></msup><mi>ε</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> Relation between δ m and δ pm can be written as [ 21 21 ] : <math display="block" id="M8"><mrow><msub><mi>δ</mi><mtext>m</mtext></msub><mo>=</mo><mtext> </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>1.26</mn><mi>r</mi></mrow><mo>)</mo></mrow><msub><mi>δ</mi><mrow><mtext>pm</mtext></mrow></msub><mo>.</mo></mrow></math> For a given material, r is a constant [ 22 22 ] . The δ m can be obtained by combining Eqs. (7) and (8). <math display="block" id="M9"><mrow><msub><mi>δ</mi><mtext>m</mtext></msub><mo>=</mo><mtext> </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>1.26</mn><mi>r</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>e</mi><mrow><mn>2.3</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>B</mi><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>9</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><mi>α</mi><msup><mi>e</mi><mrow><mn>2.3</mn></mrow></msup><mi>ε</mi></mrow><mo>)</mo></mrow></mrow></math> 2.2 Formula for δ 10-100 2 When primary electrons of 10 keV≤ W p0 ≤100 keV enter an insulator or semiconductor, R 10-100 can be expressed as [ 20 20 ] <math display="block" id="M10"><mrow><msub><mi>R</mi><mrow><mn>10</mn><mo>−</mo><mn>100</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>3.02</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup><mi>A</mi><mtext>α</mtext><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>8</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math> 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) <math display="block" id="M11"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>8</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow><mrow><mn>5.03</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mtext>p</mtext><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> 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/cm 3 ,and A α =28.1) [ 13 13 ] at W p0 =10 keV, we have R 10 =16055Å and T = 41–54Å [ 23 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 <math display="block" id="M12"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>8</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow><mrow><mn>5.03</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> The δ p in energy range of 10 keV≤ W p0 ≤100 keV can be obtained by combining Eqs. (3) and (12) <math display="block" id="M13"><mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>=</mo><mfrac><mi>B</mi><mi>ε</mi></mfrac><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>8</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow><mrow><mn>5.03</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup><msub><mi>A</mi><mi>α</mi></msub><msubsup><mi>W</mi><mrow><mi>p</mi><mn>0</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mi>R</mi></munderover><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mrow></mstyle><mo>.</mo></mrow></math> The R 10-100 is much larger than T . The internal secondary electrons excited outside T cannot be emitted into vacuum [ 23 23 ] , and T approximately equals 5/ α [ 23 23 ] . Thus, the definite integral [0, R ] of Eq.(13) can be replaced with [0,5/ α ]. Then, we have: <math display="block" id="M14"><mrow><mi>δ</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mn>5</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mi>e</mi><mn>5</mn></msup></mrow></mfrac><mfrac><mi>B</mi><mrow><mi>α</mi><mi>ε</mi></mrow></mfrac><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>8</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow><mrow><mn>5.03</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup><msub><mi>A</mi><mi>α</mi></msub><msubsup><mi>W</mi><mrow><mi>p</mi><mn>0</mn></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> The δ is composed of δ p and yield due to backscattered electrons δ r , i.e. [ 24 24 ] <math display="block" id="M15"><mrow><mi>δ</mi><mo>=</mo><msub><mi>δ</mi><mtext>p</mtext></msub><mo>+</mo><mi>η</mi><msub><mi>δ</mi><mtext>r</mtext></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>β</mi><mi>η</mi><mo stretchy="false">)</mo><msub><mi>δ</mi><mtext>p</mtext></msub><mo>.</mo></mrow></math> 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 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 25 - 28 28 25-28 ] . Because the average emission angle and mean energy of backscattered electrons from insulator or semiconductor are similar to those from metals [ 28 28 - 29 29 28-29 ] , we assume β 10-100insulator ≈2 for 10 keV≤ W p0 ≤100 keV. Therefore, we have: <math display="block" id="M16"><mrow><msub><mi>δ</mi><mrow><mn>10</mn><mtext>-</mtext><mn>100</mn></mrow></msub><mo>=</mo><mtext> </mtext><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>r</mi></mrow><mo>)</mo></mrow><msub><mi>δ</mi><mi>p</mi></msub><mo>.</mo></mrow></math> 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) <math display="block" id="M17"><mrow><msub><mi>δ</mi><mrow><mn>10</mn><mo>−</mo><mn>100</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mn>3.9</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>r</mi><mo stretchy="false">)</mo><msup><mi>Z</mi><mrow><mn>2</mn><mo>/</mo><mn>9</mn></mrow></msup><msub><mi>δ</mi><mtext>m</mtext></msub></mrow><mrow><mo stretchy="false">(</mo><mn>1.1.26</mn><mi>r</mi><mo stretchy="false">)</mo><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> 2.3 Formula for δ 2-10 2 When primary electrons of 2 keV≤ W p0 < 10 keV enter an insulator or semiconductor, R 2-10 can be expressed as [ 20 20 ] <math display="block" id="M18"><mrow><msub><mi>R</mi><mrow><mn>2</mn><mo>−</mo><mn>10</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1.03</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mi>α</mi></msub><msubsup><mi>W</mi><mrow><mi>p</mi><mn>0</mn></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math> 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) <math display="block" id="M19"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow><mrow><mn>1.545</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mtext>p</mtext><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> 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/cm 3 , A α =28.1) [ 13 13 ] , at W p0 =2000 eV, calculated with Eq. (18) we have R 2 =1222 Å and T =41–54Å [ 23 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: <math display="block" id="M20"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>W</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>S</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow><mrow><mn>1.545</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> Based on Eqs. (3) and (20), δ p in energy range of 2 keV≤ W p0 < 10 keV can be written as <math display="block" id="M21"><mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>=</mo><mfrac><mi>B</mi><mi>ε</mi></mfrac><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow><mrow><mn>1.545</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mi>R</mi></munderover><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>α</mi><mi>x</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>x</mi></mrow></mrow></mstyle><mo>.</mo></mrow></math> 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 23 ] , and T ≈5/ α [ 23 23 ] . Thus, the definite integral [0, R ] of Eq.(21) can be replaced with [0, 5/ α ], i.e., <math display="block" id="M22"><mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mn>5</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mi>e</mi><mn>5</mn></msup></mrow></mfrac><mfrac><mi>B</mi><mrow><mi>α</mi><mi>ε</mi></mrow></mfrac><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></mrow><mrow><mn>1.545</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> From Eqs. (8) and (15), we have δ p / δ r = 1/(1.26 r ) for W p0m ≤800 eV, and from Eqs. (15) and (16), we have δ p / δ r = 1/(2 r ) for 10 keV≤ W p0 ≤100 keV. There is the tendency that η increases with W p0 for W p0 ≤10 keV [ 17 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/(1.5 r ) for 2 keV≤ W p0 < 10 keV. Then, approximately, the δ for 2 keV≤ W p0 < 10 keV can be expressed as: <math display="block" id="M23"><mrow><mi>δ</mi><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>1.5</mn><mi>r</mi></mrow><mo>)</mo></mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>.</mo></mrow></math> Combining Eqs. (22) and (23), we have: <math display="block" id="M24"><mrow><msub><mi>δ</mi><mtext>p</mtext></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mn>5</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mi>e</mi><mn>5</mn></msup></mrow></mfrac><mfrac><mi>B</mi><mrow><mi>α</mi><mi>ε</mi></mrow></mfrac><mfrac><mrow><mi>ρ</mi><msup><mi>Z</mi><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1.5</mn><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1.545</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><msub><mi>A</mi><mtext>α</mtext></msub><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> From Eqs. (9) and (24), the δ for 2 keV≤ W p0 < 10 keV and W p0m ≤800 eV can be expressed as <math display="block" id="M25"><mrow><msub><mi>δ</mi><mrow><mn>2</mn><mo>−</mo><mn>10</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>14.29</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1.5</mn><mi>r</mi><mo stretchy="false">)</mo><msup><mi>Z</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><msub><mi>δ</mi><mtext>m</mtext></msub></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1.26</mn><mi>r</mi><mo stretchy="false">)</mo><msubsup><mi>W</mi><mrow><mtext>p0</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>.</mo></mrow></math> 3 Results and discussion 1 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 4 , 30 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 31 - 32 32 31-32 ] , Z , W po and r are given in Table. 1 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 31 - 32 32 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/(1.5 r ) is reasonable. 1 1 Materials [ 17 17 , 21 21 ]   1 1 δ m   1 1 W po (keV) 1 1 Calculated 1 1 Measured 1 1 Si ( Z =14, r =0.22) 1 1 1.594 [ 31 31 ]   1 1 3.0 [ 31 31 ]   1 1 0.6723 1 1 0.769 [ 31 31 ]   1 1   1 1   1 1 5.0 [ 31 31 ]   1 1 0.5207 1 1 0.49 [ 31 31 ]   1 1   1 1   1 1 10 [ 31 31 ]   1 1 0.3056 1 1 0.3431 [ 31 31 ]   1 1   1 1   1 1 20 [ 31 31 ]   1 1 0.1925 1 1 0.216 [ 31 31 ]   1 1   1 1   1 1 30 [ 31 31 ]   1 1 0.1439 1 1 0.138 [ 31 31 ]   1 1 Te ( Z =52, r =0.41) 1 1 1.63 [ 31 31 ]   1 1 3.0 [ 31 31 ]   1 1 0.875 1 1 0.891 [ 31 31 ]   1 1   1 1   1 1 5.0 [ 31 31 ]   1 1 0.6777 1 1 0.706 [ 31 31 ]   1 1   1 1   1 1 10 [ 31 31 ]   1 1 0.4451 1 1 0.376 [ 31 31 ]   1 1   1 1   1 1 20 [ 31 31 ]   1 1 0.2804 1 1 0.319 [ 31 31 ]   1 1   1 1   1 1 30 [ 31 31 ]   1 1 0.214 1 1 0.174 [ 31 31 ]   1 1   1 1 0.819 [ 11 11 ]   1 1 2.0 [ 11 11 ]   1 1 0.5384 1 1 0.495 [ 11 11 ]   1 1   1 1   1 1 2.4 [ 11 11 ]   1 1 0.4915 1 1 0.418 [ 11 11 ]   1 1 Ge ( Z =32, r =0.35) 1 1 1.142 [ 33 33 ]   1 1 2.0 [ 33 33 ]   1 1 0.688 1 1 0.696 [ 33 33 ]   1 1   1 1   1 1 2.5 [ 33 33 ]   1 1 0.6154 1 1 0.54 [ 33 33 ]   1 1   1 1   1 1 3.0 [ 33 33 ]   1 1 0.5618 1 1 0.53 [ 33 33 ]   1 1   1 1   1 1 3.5 [ 33 33 ]   1 1 0.52 1 1 0.49 [ 33 33 ]   1 1   1 1   1 1 4.0 [ 33 33 ]   1 1 0.4865 1 1 0.45 [ 33 33 ]   1 1   1 1   1 1 4.5 [ 33 33 ]   1 1 0.4587 1 1 0.42 [ 33 33 ]   1 1   1 1   1 1 5.0 [ 33 33 ]   1 1 0.4352 1 1 0.42 [ 33 33 ]   1 1   1 1 0.743 [ 11 11 ]   1 1 2.0 [ 11 11 ]   1 1 0.4477 1 1 0.398 [ 11 11 ]   1 1   1 1   1 1 3.0 [ 11 11 ]   1 1 0.3655 1 1 0.312 [ 11 11 ]   1 1   1 1   1 1 4.0 [ 11 11 ]   1 1 0.3165 1 1 0.264 [ 11 11 ]   1 1   1 1 1.62 [ 31 31 ]   1 1 5.0 [ 31 31 ]   1 1 0.6174 1 1 0.649 [ 31 31 ]   1 1   1 1   1 1 20.0 [ 31 31 ]   1 1 0.2458 1 1 0.228 [ 31 31 ]   1 1 C ( Z =6, r =0.1) 1 1 1.56 [ 31 31 ]   1 1 5.0 [ 31 31 ]   1 1 0.434 1 1 0.501 [ 31 31 ]   1 1   1 1   1 1 20 [ 31 31 ]   1 1 0.1475 1 1 0.125 [ 31 31 ]   1 1   1 1 1.136 [ 33 33 ]   1 1 2.0 [ 34 34 ]   1 1 0.5 1 1 0.527 [ 34 34 ]   1 1   1 1   1 1 2.5 [ 34 34 ]   1 1 0.447 1 1 0.442 [ 34 34 ]   1 1   1 1   1 1 3.0 [ 34 34 ]   1 1 0.408 1 1 0.4 [ 34 34 ]   1 1   1 1   1 1 4.0 [ 34 34 ]   1 1 0.3533 1 1 0.316 [ 34 34 ]   1 1   1 1   1 1 5.0 [ 34 34 ]   1 1 0.316 1 1 0.315 [ 34 34 ]   1 1 Indium tin oxide ( Z =35.7, r =0.36) 1 1 2.52 [ 32 32 ]   1 1 2.0 [ 32 32 ]   1 1 1.5477 1 1 1.55 [ 31 31 ]   1 1   1 1   1 1 3.0 [ 32 32 ]   1 1 1.2637 1 1 1.31 [ 32 32 ]   1 1   1 1   1 1 5.0 [ 32 32 ]   1 1 0.9789 1 1 1.03 [ 32 32 ]   1 1   1 1   1 1 7.0 [ 32 32 ]   1 1 0.8273 1 1 0.72 [ 32 32 ]   1 1   1 1   1 1 10 [ 32 32 ]   1 1 0.6356 1 1 0.73 [ 32 32 ]   1 1 Indium zinc oxide ( Z =29, r =0.34) 1 1 2.67 [ 32 32 ]   1 1 2.0 [ 32 32 ]   1 1 1.581 1 1 1.61 [ 32 32 ]   1 1   1 1   1 1 3.0 [ 32 32 ]   1 1 1.291 1 1 1.31 [ 32 32 ]   1 1   1 1   1 1 5.0 [ 32 32 ]   1 1 1.0 1 1 1.02 [ 32 32 ]   1 1   1 1   1 1 7.0 [ 32 32 ]   1 1 0.845 1 1 0.86 [ 32 32 ]   1 1   1 1   1 1 10 [ 32 32 ]   1 1 0.6278 1 1 0.72 [ 32 32 ]   1 1 Al 2 O 3 ( Z =10, r =0.18) 1 1 6.23 [ 35 35 ]   1 1 2.0 [ 35 35 ]   1 1 3.025 1 1 3.83 [ 35 35 ]   1 1   1 1   1 1 3.0 [ 35 35 ]   1 1 2.47 1 1 2.44 [ 35 35 ]   1 1 V 2 O 5 ( Z =12.3, r =0.2) 1 1 1.216 [ 36 36 ]   1 1 2.01 [ 36 36 ]   1 1 0.6114 1 1 0.718 [ 36 36 ]   1 1   1 1   1 1 2.54 [ 36 36 ]   1 1 0.5439 1 1 0.669 [ 36 36 ]   1 1   1 1   1 1 3.044 [ 36 36 ]   1 1 0.4968 1 1 0.578 [ 36 36 ]   1 1   1 1   1 1 3.572 [ 36 36 ]   1 1 0.4586 1 1 0.53 [ 36 36 ]   1 1   1 1   1 1 4.034 [ 36 36 ]   1 1 0.4316 1 1 0.487 [ 36 36 ]   1 1   1 1   1 1 5.056 [ 36 36 ]   1 1 0.3855 1 1 0.389 [ 36 36 ]   1 1   1 1   1 1 5.571 [ 36 36 ]   1 1 0.367 1 1 0.347 [ 36 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 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 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 1 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. 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Phys . 51 , 3441 - 3443 ( 1980 ). doi: 10.1063/1.326320 http://doi.org/10.1063/1.326320 Enhanced secondary electron emission yield from doped vanadate-phosphate glasses A. van der Ziel , Solid State Physical Electronics , 2 nd ed ( Englewood Cliffs : Prentice-Hall ) p 172 ( 1968 ). Solid State Physical Electronics 10.1007/s41365-017-0291-y This work was supported by the National Natural Science Foundation of China (No.11473049). 2016-07-18 2017-03-16 2017-04-22 2017-09-06 2017-10-01 2022-02-23 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0291-y http://dx.doi.org/10.1007/s41365-017-0291-y . 2017 Nuclear Science and Techniques 10 28