劉亦凡，謝愛根，董洪杰

(南京信息工程大學 物理與光電工程學院 ，南京 210044)

0 Introduction

A negative electron affinity (NEA) semiconductor means that the vacuum level of the semiconductor exists below conduction band minimum at the surface, which is a very rare property. Under NEA, internal secondary electrons in the conduction band can easily emit from the surface as there is no barrier at the semiconductor surface[1].The secondary electron yield(SEY)δof NEA semiconductors such as Si and GaAs in general far exceed those of positive electron affinity emitters because NEA semiconductors have much larger mean escape depth of secondary electron 1/α[2]. Thus, NEA semiconductors are outstanding secondary electron emitters and are applied in current amplifiers, vacuum tube applications, electronic information technology,etc[1, 3-4]. Therefore, NEA semiconductor is a very important topic[5-8].

Due to different bulk properties such as dopant type and doping concentration and surface terminations such as the type of adsorbate, the extent of the adsorbate coverage and the presence of coad-sorbed molecules, some NEA semiconductors such as NEA diamond[4]and GaAs[9]exhibit very high, but widely varying,δand maximum SEYδm. Thus, bulk properties and surface terminations of a given NEA semiconductor decideδandδm. According to the expressions ofδm[10-11]and the fact that theδof given emitter and incident energy of primary electronEpois proportional to itsδm, it is known that theBand 1/αof a given kind of semiconductor almost decide the values ofδat givenEpoandδm,Bis the probability that an internal secondary electron escapes into vacuum upon reaching the surface of emitter. Thus, from the fact that sample preparations decide bulk properties and surface terminations of a given NEA semiconductor[4], it is known that sample preparations of a given NEA semiconductor decide theδat givenEpo,δm,Band 1/α. TheBis inaccessible to measurement, and it is very difficult to measure 1/α. Therefore, from the fact that theBand 1/αof a given kind of semiconductor almost decide the value ofδmand theδat givenEpo, it concludes that the theoretical researches ofBand 1/αare necessary and help to research quantitative influences of different sample preparations on parameters of SEE such asδat givenEpo,δm,Band 1/α. Hence, from the relationships amongδm,δ,Band 1/αand quantitative influences of different sample preparations on parameters of SEE obtained by the theoretical researches ofBand 1/α, we can change the sample preparations and produce desirable NEA emitter such as NEA diamond.In other words, the theoretical researches ofBand 1/αhelp to produce desirable NEA emitters such as NEA diamond and GaAs those exhibit very high, but widely varying,δandδmbecause of different sample preparations.

According to the characteristics of SEE from NEA semiconductors with 0.8keV≤Epomax≤5keV,R, existing universal formulas forδof NEA semiconductors[12]and experimental data[4,13-14], special formulas forδat 0.5Epomax≤Epo≤10Epomaxof NEA diamond and GaN with 2keV≤Epomax≤5keV andδat 0.8keV≤Epo≤3keV of NEA diamond and GaN with 0.8keV≤Epomax≤2keV were deduced and experimentally proved, respectively; whereRis primary range,Epomaxis theEpoat whichδreachesδm.The formula forBof NEA semiconductors with 0.8keV≤Epomax≤5keV deduced in this study could be used to calculateB, and the method presented here of calculating the 1/αof NEA semiconductors with 0.8keV≤Epomax≤5keV is correct. Thus, according to the fact that the theoretical researches ofBand 1/αhelp to research quantitative influences of different sample preparations on parameters of SEE and produce the desirable NEA emitters, it concludes that this study’s research onBand 1/αhelp to research quantitative influences of different sample preparations on SEE from NEA semiconductors and produce desirable NEA emitters such as NEA diamond.

HighδNEA diamond is very valuable for electron multiplication in devices such as crossed-field amplifiers and electron multipliers[4].Thus, NEA diamond is an important topic[8, 15-18]. Therefore, this study focuses on NEA diamond. Of course, the method presented here of researchingδ,Band 1/αof NEA diamond and GaN can be used to researchδ,Band 1/αof NEA semiconductor with 0.8keV≤Epomax≤5keV.

1 Universal formulas for R and secondary electron yield

1.1 Primary range

According to theR-Eporelationship deduced from the power potential law, the relation amongR, the energy exponentQandEpois expressed as[19]: whereQis a constant in the sameEporange[19], andAdepends on the atomic weightAα, material densityρand atomic numberZin the sameEporange[19]. When primary electrons at 0.8keV≤Epo≤2keV enter a secondary electron emitter,theRat 0.8keV≤Epo≤2keV can be expressed as[19]

(1)

When primary electrons at 2keV≤Epo≤10keV enter a secondary electron emitter, theRat 2keV≤Epo≤10keV can be expressed in terms ofρ,Z,Aα,Epo[19]

(2)

When primary electrons at 10keV≤Epo≤100keV enter a secondary electron emitter, theRat 10keV≤Epo≤100keV can be expressed in terms ofρ,Z,Aα,Epo[19]

(3)

1.2 Universal formula for δ

The universal formula forδat 0.1keV≤Epo≤10keV of NEA semiconductors can be expressed as[12]:

(4)

whereεis the average energy required to produce an internal secondary electron in a semiconductor,αis the absorption coefficient,R0.1-10keVisRat 0.1keV≤Epo≤10keV, the factorK(Epo,ρ,Z) of given NEA semiconductor andEpois approximately equal to a constant and less than 1,ris the high energy back-scattering coefficient which is nearly independent ofEpoand can be approximately expressed by[20]

r=-0.0254+0.016Z-1.86×10-4Z2+8.3×10-7Z3

(5)

The universal formula forδat 10keV≤Ep≤100keV of NEA semiconductors can be expressed as[12]:

(6)

2 SEE from NEA GaN with Epomax=3.0keV

The ratio ofREpomaxin the NEA semiconductors to the corresponding 1/αcan be expressed as[12]:

(7)

wherenisαconstant forαgiven NEA semiconductor,REpomaxisRatEpomax.

Seen from Fig.1, it is known that theEpomaxof NEA GaN withEpomax=3.0keV is 3.0keV[13].R3.0 keVcalculated with Eq.(2) and parameters of GaN[13, 21](ρ=6.1g/cm3,Aα=42,Z=19,Epo=3.0keV) is equal to 1001.972 ?. Therefore, from Eq.(7), the (1/α) of NEA GaN withEpomax=3.0keV can be expressed as:

Fig.1 Comparison between experimental δ of NEA GaN[13] with Epomax=3.0keV and corresponding calculated ones

(8)

Therof GaN calculated with Eq.(5) andZ=19 is equal to 0.206. As seen from Fig. 1, it is known that theEpomaxof NEA GaN withEpomax=3.0keV is in the range of 2keV≤Epomax≤5keV. According to characteristics of SEE, the course of deducing Eq.(10) of former study[12]and the conclusion thatK(Epo,ρ,Z) decreases with increasingEpoin the range ofEpo≥100 eV[12], we assumed thatK(Epo,ρ,Z)of the NEA semiconductors with 2keV≤Epomax≤5keV decreases slowly with increasingEpoin the range of 0.5Epomax≤Epo≤10Epomax, and thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5[12]. Thus, from the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5,we take theK(Epo,ρ,Z=19) at 0.5Epomax≤Epo≤10Epomaxof the NEA GaN withEpomax=3.0keV to be a constantK(Epo,ρ,Z=19)C3; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA GaN[13,21](ρ=6.1g/cm3,Aα=42,Z=19,r=0.206,Epomax=3.0keV), the assumption thatK(Epo,ρ,Z=19) at 0.5Epmax≤Epo≤10Epomaxof the NEA GaN withEpomax=3.0keV equalsK(Epo,ρ,Z=19)C3and Eqs.(2), (4) and (8), theδat 2keV≤Epo≤10keV of the NEA GaN withEpomax=3.0keV can be expressed as follows:

(9)

Eq.(9), the result thatnof Eq.(9) approximately equals 2.2649 is obtained. Therefore, the (1/α) of NEA GaN withEpomax=3.0keV calculated with Eq.(8) andn=2.2649 is equal to 442.39 ?. Based on the relation between experimentalδ3.0 keVof the NEA GaN withEpomax=3.0keV equaling 51[13]and theδ3.0 keVcalculated with Eq.(9),Epo=3.0keV andn=2.2649 equaling 1.3024×103[BK(Epo,ρ,Z=19)C3]/ε, [BK(Epo,ρ,Z=19)C3]/εequaling 3.916×10-2is obtained; according to the relation between the experimentalδ5.0 keVof the NEA GaN withEpomax=3.0keV equaling 49[13]and theδ5.0 keVcalculated with Eq.(9),Epo=5.0keV andn=2.2649 equaling 1.2×103[BK(Epo,ρ,Z=19)C3]/ε, [BK(Epo,ρ,Z=19)C3]/εequaling 4.083×10-2is obtained; on the basis of the relation between the experimentalδ7.0 keVof the NEA GaN withEpomax=3.0keV equaling 45[13]and the calculatedδ7.0 keVcalculated with Eq.(9),Epo=7.0keV andn=2.2649 equaling 1.097561×103[BK(Epo,ρ,Z=19)C3]/ε, [BK(Epo,ρ,Z=19)C3]/εequaling 4.1×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=19)C3]/εequaling 4.033×10-2is obtained.

From the assumption thatK(Epo,ρ,Z=19) at 0.5Epomax≤Epo≤10Epomaxof NEA GaN equalsK(Epo,ρ,Z=19)C3, parameters[13, 21](ρ=6.1g/cm3,Aα=42,Z=19, 1/α= 442.39 ?,r=0.206,K(Epo,ρ,Z=19)C3(B/ε)=4.033×10-2,Epomax=3.0keV) and Eqs.(3) and (6), theδat 10keV≤Epo≤30keV of NEA GaN withEpomax=3.0keV can be expressed as:

(10)

According to the parameters of NEA GaN withEpomax=3.0keV[13](n=2.2649,K(Epo,ρ,Z=19)C3(B/ε) =4.033×10-2) and Eq.(9), theδat 2keV≤Epo≤10keV of NEA GaN withEpomax=3.0keV can be expressed as:

(11)

From the assumption thatK(Epo,ρ,Z=19) at 0.5Epomax≤Epo≤10Epomaxof NEA GaN equalsK(Epo,ρ,Z=19)C3, parameters[13, 21](ρ=6.1g/cm3,Aα=42,Z=19, 1/α= 442.39 ?,r=0.206,K(Epo,ρ,Z=19)C3(B/ε)=4.033×10-2,Epomax=3.0keV) and Eqs.(1) and (4), theδat 1.5keV≤Epo≤2keV of NEA GaN withEpomax=3.0keV can be expressed as:

(12)

3 SEE from NEA diamond with Epomax=2.75keV

Seen from Fig. 2, it is known that theEpomaxof NEA diamond withEpomax=2.75keV is 2.75keV[4].R2.75 keVcalculated with Eq.(2) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=2.75keV) is equal to 1137.68 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=2.75 keV can be expressed as:

Fig.2 Comparison between experimental δ of NEA diamond with Epomax=2.75keV and diamond with Epomax=1.72keV[4] and corresponding calculated ones

(13)

The 2rof diamond calculated with Eq.(5) andZ=6 is equal to 0.128. Seen from Fig. 2, it is known that theEpomaxof NEA diamond withEpomax=2.75keV[4]is in the range of 2keV≤Epomax≤5keV. Thus, from the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5, we take theK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.75keV to be a constantK(Epo,ρ,Z=6)C2.75; and the ratio ofBtoεis independent ofEpo[22-24].Therefore, from parameters of NEA diamond withEpomax=2.75keV[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 2r=0.128,Epomax=2.75keV), the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.75keV equalsK(Epo,ρ,Z=6)C2.75and Eqs.(2), (4) and (13), theδat 2.0keV≤Epo≤10keV of the NEA diamond withEpomax=2.75keV can be expressed as follows:

(14)

Theδat 2.5keV≤Epo≤10keV of the NEA diamond withEpomax=2.75keV reachesδmatEpo=2.75keV. Thus, from Eq.(14), the result that thenof Eq.(14) approximately equals 2.0043 is obtained. Therefore, the 1/αof NEA diamond withEpomax=2.75keV calculated with Eq.(13) andn=2.0043 is equal to 567.62 ?. Based on the relation between the experimentalδ2.75 keVof the NEA diamond withEpomax=2.75keV equaling 18.5[4]and theδ2.75 keVcalculated with Eq.(14),Epo=2.75keV andn=2.0043 equaling 1.219357×103[BK(Epo,ρ,Z=6)C2.75]/ε, [BK(Epo,ρ,Z=6)C2.75]/εequaling 1.517×10-2is obtained; on the basis of the relation between the experimentalδ2 keVof the NEA diamond withEpomax=2.75keV equaling 16[4]and theδ2 keVcalculated with Eq.(14),Epo=2keV andn=2.0043 equaling 1.16588268×103[BK(Epo,ρ,Z=6)C2.75]/ε, [BK(Epo,ρ,Z=6)C2.75]/εequaling 1.372×10-2is obtained;according to the relation between the experimentalδ2.85 keVof the NEA diamond withEpomax=2.75keV equaling 18.2[4]and theδ2.85 keVcalculated with Eq.(14),Epo=2.85keV andn=2.0043 equaling 1.219×103[BK(Epo,ρ,Z=6)C2.75]/ε, [BK(Epo,ρ,Z=6)C2.75]/εequaling 1.493×10-2is obtained; according to the relation between the experimentalδ2.2 keVof the NEA diamond withEpomax=2.75keV equaling 17.1[4]and theδ2.2 keVcalculated with Eq.(14),Epo=2.2keV andn=2.0043 equaling 1.1926579×103[BK(Epo,ρ,Z=6)C2.75]/ε, [BK(Epo,ρ,Z=6)C2.75]/εequaling 1.434×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C2.75]/εequaling 1.45×10-2is obtained.

According to the parameters of NEA diamond withEpomax=2.75keV (n=2.0043, [BK(Epo,ρ,Z=6)C2.75]/ε=1.45×10-2) and Eq.(14), theδat 2keV≤Epo≤10keV of NEA diamond withEpomax=2.75keV can be expressed as:

(15)

From the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof NEA diamond withEpomax=2.75keV equalsK(Epo,ρ,Z=6)C2.75, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α=567.62 ?, 2r=0.128), [BK(Epo,ρ,Z=6)C2.75]/ε=1.45×10-2,Epomax=2.75keV and Eqs.(1) and (4), theδat 1.375keV≤Epo≤2keV of NEA diamond withEpomax=2.75keV can be expressed as:

(16)

4 SEE from NEA diamond with Epomax=2.64keV

Seen from Fig. 3, it is known that theEpomaxof NEA diamond withEpomax=2.64keV is 2.64keV[4].R2.64 keVcalculated with Eq.(2) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=2.64keV) is equal to 1070.1 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=2.64keV can be expressed as:

(17)

Seen from Fig. 3, it is known that theEpomaxof NEA diamond withEpomax=2.64keV is in the range of 2keV≤Epomax≤5keV. Thus, from the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5, we take theK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.64keV to be a constantK(Epo,ρ,Z=6)C2.64; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA diamond withEpomax=2.64keV[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 2r=0.128,Epomax=2.64keV), the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.64keV equalsK(Epo,ρ,Z=6)C2.64and Eqs.(2), (4) and (17), theδat 2.0keV≤Epo≤10keV of the NEA diamond withEpomax=2.64keV can be expressed as follows:

Fig.3 Comparison between experimental δ of NEA diamond with Epomax=2.64keV and diamond with Epomax=2.3keV[4] and corresponding calculated ones

(18)

Theδat 2keV≤Epo≤10keV of the NEA diamond withEpomax=2.64keV reachesδmatEpo=2.64keV. Thus, from Eq.(18), the result that thenof Eq.(18) approximately equals 1.9994 is obtained. Therefore, the (1/α) of NEA diamond withEpomax=2.64keV calculated with Eq.(17) andn=1.9994 is equal to 535.21 ?. Based on the relation between the experimentalδ2.64 keVof the NEA diamond withEpomax=2.64keV equaling 24.7[4]and theδ2.64 keVcalculated with Eq.(18),Epo=2.64keV andn=1.9994 equaling 1.1711×103[BK(Epo,ρ,Z=6)C2.64]/ε, [BK(Epo,ρ,Z=6)C2.64]/εequaling 2.109×10-2is obtained；on the basis of the relation between the experimentalδ2.9 keVof the NEA diamond withEpomax=2.64keV equaling 23[4]and theδ2.9 keVcalculated with Eq.(18),Epo=2.9keV andn=1.9994 equaling 1.166656×103[BK(Epo,ρ,Z=6)C2.64]/ε, [BK(Epo,ρ,Z=6)C2.64]/εequaling 1.9716×10-2is obtained;according to the relation between the experimentalδ2.5 keVof the NEA diamond withEpomax=2.64keV equaling 24[4]and theδ2.5 keVcalculated with Eq.(18),Epo=2.5keV andn=1.9994 equaling 1.1695385×103[BK(Epo,ρ,Z=6)C2.64]/ε, [BK(Epo,ρ,Z=6)C2.64]/εequaling 2.052 1×10-2is obtained; according to the relation between the experimentalδ2.2 keVof the NEA diamond withEpomax=2.64keV equaling 22.5[4]and theδ2.2 keVcalculated with Eq.(18),Epo=2.2keV andn=1.9994 equaling 1.15383×103[BK(Epo,ρ,Z=6)C2.64]/ε, [BK(Epo,ρ,Z=6)C2.64]/εequaling 1.495×10-2is obtained；on the basis of the relation between the experimentalδ2.1 keVof the NEA diamond withEpomax=2.64keV equaling 21.4[4]and theδ2.1 keVcalculated with Eq.(18),Epo=2.1keV andn=1.9994 equaling 1.144×103[BK(Epo,ρ,Z=6)C2.64]/ε, [BK(Epo,ρ,Z=6)C2.64]/εequaling 1.87×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C2.64]/εequaling 1.991×10-2is obtained.

According to the parameters of NEA diamond withEpomax=2.64keV (n=1.9994, [BK(Epo,ρ,Z=6)C2.64]/ε=1.991×10-2) and Eq.(18), theδat 2keV≤Epo≤10keV of NEA diamond withEpomax=2.64keV can be expressed as:

(19)

From the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof NEA diamond withEpomax=2.64keV equalsK(Epo,ρ,Z=6)C2.64, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α=535.21 ?, 2r=0.128), [BK(Epo,ρ,Z=6)C2.64]/ε=1.991×10-2,Epomax=2.64keV and Eqs.(1) and (4), theδat 1.32keV≤Epo≤2keV of NEA diamond withEpomax=2.64keV can be expressed as:

(20)

5 SEE from NEA diamond with Epomax=2.3keV

Seen from Fig. 3, it is known that theEpomaxof NEA diamond withEpomax=2.3keV is 2.3keV[4].R2.3 keVcalculated with Eq.(2) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=2.3keV) is equal to 870.18 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=2.3 keV can be expressed as:

(21)

Seen from Fig. 3, it is known that theEpomaxof NEA diamond withEpomax=2.3keV is in the range of 2keV≤Epomax≤5keV. Thus, from the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5, we take theK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.3keV to be a constantK(Epo,ρ,Z=6)C2.3; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA diamond withEpomax=2.3keV[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 2r=0.128,Epomax=2.3keV), the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof the NEA diamond withEpomax=2.3keV equalsK(Epo,ρ,Z=6)C2.3and Eqs.(2), (4) and (21), theδat 2.0keV≤Epo≤10keV of the NEA diamond withEpomax=2.3keV can be expressed as follows:

(22)

Theδat 2keV≤Epo≤10keV of the NEA diamond withEpomax=2.3keV reachesδmatEpo=2.3keV. Thus, from Eq.(22), the result that thenof Eq.(22) approximately equals 1.9849 is obtained. Therefore, the (1/α) of NEA diamond withEpomax=2.3keV calculated with Eq.(21) andn=1.9849 is equal to 438.4 ?. Based on the relation between the experimentalδ2.3 keVof the NEA diamond withEpomax=2.3keV equaling 30[4]and theδ2.3 keVcalculated with Eq.(22),Epo=2.3keV andn=1.9849 equaling 1.02146×103[BK(Epo,ρ,Z=6)C2.3]/ε, [BK(Epo,ρ,Z=6)C2.3]/εequaling 2.937×10-2is obtained; on the basis of relation between the experimentalδ2.2 keVof the NEA diamond withEpomax=2.3keV equaling 29.9[4]and theδ2.2 keVcalculated with Eq.(22),Epo=2.2keV andn=1.9849 equaling 1.0205×103[BK(Epo,ρ,Z=6)C2.3]/ε, [BK(Epo,ρ,Z=6)C2.3]/εequaling 2.93×10-2is obtained; on the basis of the relation between the experimentalδ2.1 keVof the NEA diamond withEpomax=2.3keV equaling 29.8[4]and theδ2.1 keVcalculated with Eq.(22),Epo=2.1keV andn=1.9849 equaling 1.017619×103[BK(Epo,ρ,Z=6)C2.3]/ε, [BK(Epo,ρ,Z=6)C2.3]/εequaling 2.928×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C2.3]/εequaling 2.93×10-2is obtained.

According to the parameters of NEA diamond withEpomax=2.3keV (n=1.9849, [BK(Epo,ρ,Z=6)C2.3]/ε=2.93×10-2) and Eq.(22), theδat 2keV≤Epo≤10keV of NEA diamond withEpomax=2.3keV can be expressed as:

(23)

From the assumption thatK(Epo,ρ,Z=6) at 0.5Epomax≤Epo≤10Epomaxof NEA diamond withEpomax=2.3keV equalsK(Epo,ρ,Z=6)C2.3, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α=438.4 ?, 2r=0.128), [BK(Epo,ρ,Z=6)C2.3]/ε=2.93×10-2,Epomax=2.3keV) and Eqs.(1) and (4), theδat 1.15keV≤Epo≤2keV of NEA diamond withEpomax=2.3keV can be expressed as:

(24)

6 SEE from NEA GaN with Epomax=1.0keV

Seen from Fig.4, it is known that theEpomaxof NEA GaN withEpomax=1.0keV is 1.0keV[14].R1.0 keVcalculated with Eq.(1) and parameters of GaN[14, 21](ρ=6.1g/cm3,Aα=42,Z=19,Epo=1.0keV) is equal to 213.3343 ?. Therefore, from Eq.(7), the (1/α) of NEA GaN withEpomax=1.0keV can be expressed as:

Fig. 4 Comparison between experimental δ of NEA diamond with Epomax=0.85keV[4] and GaN with Epomax=1.0keV[14] and corresponding calculated ones

(25)

Seen from Fig. 4, it is known that theEpomaxof NEA GaN withEpomax=1.0keV is in the range of 0.8keV≤Epomax≤2keV. According to the conclusion thatK(Epo,ρ,Z)decreases with increasingEpoin the range ofEpo≥100eV, we assumed thatK(Epo,ρ,Z)of the NEA semiconductors with 0.8keV≤Epomax≤2keV decreases slowly with increasingEpoin the range of 0.8keV≤Epo≤3keV, and thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2. Thus, from the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2, we take theK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3keV of the NEA GaN withEpomax=1.0keV to be a constantK(Epo,ρ,Z=19)C1; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA GaN[14, 21](ρ=6.1g/cm3,Aα=42,Z=19,r=0.206,Epomax=1.0keV), the assumption thatK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3 keV of the NEA GaN withEpomax=1.0keV equalsK(Epo,ρ,Z=19)C1and Eqs.(1), (4) and (25), theδat 0.8keV≤Epo≤2keV of the NEA GaN withEpomax=1.0keV can be expressed as follows:

(26)

Theδat 0.8keV≤Epo≤2keV of the NEA GaN reaches itsδmatEpo=1.0keV. Thus, from Eq.(26), the result thatnof Eq.(26) approximately equals 2.4766 is obtained. Therefore, the (1/α) of NEA GaN withEpomax=1.0keV calculated with Eq.(25) andn=2.4766 is equal to 86.14 ?. Based on the relation between experimentalδ1.0 keVof the NEA GaN withEpomax=1.0keV equaling 6.1[14]and theδ1.0 keVcalculated with Eq.(26),Epo=1.0keV andn=2.4766 equaling 3.7946×102[BK(Epo,ρ,Z=19)C1]/ε, [BK(Epo,ρ,Z=19)C1]/εequaling 1.6076×10-2is obtained; according to the relation between the experimentalδ0.8 keVof the NEA GaN withEpomax=1.0keV equaling 6.0[14]and theδ0.8 keVcalculated with Eq.(26),Epo=0.8keV andn=2.4766 equaling 3.731×102[BK(Epo,ρ,Z=19)C1]/ε, [BK(Epo,ρ,Z=19)C1]/εequaling 1.6082×10-2is obtained; on the basis of the relation between the experimentalδ1.5 keVof the NEA GaN withEpomax=1.0keV equaling 5.95[14]and the calculatedδ1.5 keVcalculated with Eq.(26),Epo=1.5keV andn=2.4766 equaling 3.6242×102[BK(Epo,ρ,Z=19)C1]/ε, [BK(Epo,ρ,Z=19)C1]/εequaling 1.6417×10-2is obtained; according to the relation between the experimentalδ1.75 keVof the NEA GaN withEpomax=1.0keV equaling 5.69[14]and theδ1.75 keVcalculated with Eq.(26),Epo=1.75keV andn=2.4766 equaling 3.50256×102[BK(Epo,ρ,Z=19)C1]/ε, [BK(Epo,ρ,Z=19)C1]/εequaling 1.6245×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=19)C1]/εequaling 1.62×10-2is obtained.

According to the parameters of NEA GaN withEpomax=1.0keV (n=2.4766,K(Epo,ρ,Z=19)C1(B/ε) =1.62×10-2) and Eq.(26), theδat 0.8keV≤Epo≤2keV of NEA GaN withEpomax=1.0keV can be expressed as:

(27)

From the assumption thatK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3keV of NEA GaN equalsK(Epo,ρ,Z=19)C1, parameters[14, 21](ρ=6.1g/cm3,Aα=42,Z=19, 1/α= 86.14 ?,r=0.206,K(Epo,ρ,Z=19)C1(B/ε)=1.62×10-2,Epomax=1.0keV) and Eqs.(2) and (4), theδat 2keV≤Epo≤3keV of NEA GaN withEpomax=1.0keV can be expressed as:

(28)

7 SEE from NEA GaN with Epomax=1.25keV

Seen from Fig. 5, it is known that theEpomaxof NEA GaN withEpomax=1.25keV is 1.25keV[14].R1.25 keVcalculated with Eq.(1) and parameters of GaN[14, 21](ρ=6.1g/cm3,Aα=42,Z=19,Epo=1.25keV) is equal to 287.2614 ?. Therefore, from Eq.(7), the (1/α) of NEA GaN withEpomax=1.25keV can be expressed as:

(29)

Seen from Fig. 5, it is known that theEpomaxof NEA GaN withEpomax=1.25keV is in the range of 0.8keV≤Epomax≤2keV.Thus, from the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2, we take theK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3keV of the NEA GaN withEpomax=1.25keV to be a constantK(Epo,ρ,Z=19)C1.25; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA GaN[14, 21](ρ=6.1g/cm3,Aα=42,Z=19,r=0.206,Epomax=1.25keV), the assumption thatK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3 keV of the NEA GaN withEpomax=1.25keV equalsK(Epo,ρ,Z=19)C1.25and Eqs.(1), (4) and (29), theδat 0.8keV≤Epo≤2keV of the NEA GaN withEpomax=1.25keV can be expressed as follows:

Fig. 5 Comparison between experimental δ of NEA diamond with Epomax=1.1keV[4] and GaN with Epomax=1.25keV[14] and corresponding calculated ones

(30)

Theδat 0.8keV≤Epo≤2keV of the NEA GaN reaches itsδmatEpo=1.25keV. Thus, from Eq.(30), the result thatnof Eq.(30) approximately equals 2.5205 is obtained. Therefore, the (1/α) of NEA GaN withEpomax=1.25keV calculated with Eq.(29) andn=2.5205 is equal to 113.97 ?. Based on the relation between experimentalδ1.25 keVof the NEA GaN withEpomax=1.25keV equaling 7.0[14]and theδ1.25 keVcalculated with Eq.(30),Epo=1.25keV andn=2.5205 equaling 4.7154×102[BK(Epo,ρ,Z=19)C1.25]/ε, [BK(Epo,ρ,Z=19)C1.25]/εequaling 1.4845×10-2is obtained; according to the relation between the experimentalδ0.8 keVof the NEA GaN withEpomax=1.25keV equaling 6.3[14]and theδ0.8 keVcalculated with Eq.(30),Epo=0.8keV andn=2.5205 equaling 4.40734×102[BK(Epo,ρ,Z=19)C1.25]/ε, [BK(Epo,ρ,Z=19)C1.25]/εequaling 1.4294×10-2is obtained; on the basis of the relation between the experimentalδ1 keVof the NEA GaN withEpomax=1.25keV equaling 6.9[14]and the calculatedδ1 keVcalculated with Eq.(30),Epo=1keV andn=2.5205 equaling 4.637765×102[BK(Epo,ρ,Z=19)C1.25]/ε, [BK(Epo,ρ,Z=19)C1.25]/εequaling 1.487 8×10-2is obtained; according to the relation between the experimentalδ1.75 keVof the NEA GaN withEpomax=1.25keV equaling 6.65[14]and theδ1.75 keVcalculated with Eq.(30),Epo=1.75keV andn=2.5205 equaling 4.5692×102[BK(Epo,ρ,Z=19)C1.25]/ε, [BK(Epo,ρ,Z=19)C1.25]/εequaling 1. 4554×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=19)C1.25]/εequaling1.4643×10-2is obtained.

According to the parameters of NEA GaN withEpomax=1.25keV (n=2.5205,K(Epo,ρ,Z=19)C1.25(B/ε)=1.4643×10-2) and Eq.(30), theδat 0.8keV≤Epo≤2keV of NEA GaN withEpomax=1.25keV can be expressed as:

(31)

From the assumption thatK(Epo,ρ,Z=19) at 0.8keV≤Epo≤3keV of NEA GaN equalsK(Epo,ρ,Z=19)C1.25, parameters[14, 21](ρ=6.1g/cm3,Aα=42,Z=19, 1/α= 113.97 ?,r=0.206,K(Epo,ρ,Z=19)C1.25(B/ε)=1.4643×10-2,Epomax=1.25keV) and Eqs.(2) and (4), theδat 2keV≤Epo≤3keV of NEA GaN withEpomax=1.25keV can be expressed as:

(32)

8 SEE from NEA diamond with Epomax=0.85keV

Seen from Fig. 4, it is known that theEpomaxof NEA diamond withEpomax=0.85keV is 0.85keV[4].R0.85 keVcalculated with Eq.(1) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=0.85keV) is equal to 208.4663 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=0.85keV can be expressed as:

(33)

Seen from Fig. 4, it is known that theEpomaxof NEA diamond withEpomax=0.85keV is in the range of 0.8keV≤Epomax≤2keV.Thus, from the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2.5, we take theK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of the NEA diamond withEpomax=0.85keV to be a constantK(Epo,ρ,Z=6)C0.85; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,r=0.064,Epomax=0.85keV), the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of the NEA diamond withEpomax=0.85keV equalsK(Epo,ρ,Z=6)C0.85and Eqs.(1), (4) and (33), theδat 0.8keV≤Epo≤2keV of the NEA diamond withEpomax=0.85keV can be expressed as follows:

(34)

Theδat 0.8keV≤Epo≤2keV of the NEA diamond reaches itsδmatEpo=0.85keV. Thus, from Eq.(34), the result thatnof Eq.(34) approximately equals 2.3719 is obtained. Therefore, the (1/α) of NEA diamond withEpomax=0.85keV calculated with Eq.(33) andn=2.3719 is equal to 87.89 ?. Based on the relation between experimentalδ0.85 keVof the NEA diamond withEpomax=0.85keV equaling 4.667[4]and theδ0.85 keVcalculated with Eq.(34),Epo=0.85keV andn=2.3719 equaling 3.27081×102[BK(Epo,ρ,Z=6)C0.85]/ε, [BK(Epo,ρ,Z=6)C0.85]/εequaling 1.42686×10-2is obtained; according to the relation between the experimentalδ1.8 keVof the NEA diamond withEpomax=0.85keV equaling 4.05[4]and theδ1.8 keVcalculated with Eq.(34),Epo=1.8keV andn=2.3719 equaling 2.831786 4×102[BK(Epo,ρ,Z=6)C0.85]/ε, [BK(Epo,ρ,Z=6)C0.85]/εequaling 1.43×10-2is obtained; on the basis of the relation between the experimentalδ1 keVof the NEA diamond withEpomax=0.85keV equaling 4.6[4]and the calculatedδ1 keVcalculated with Eq.(34),Epo=1keV andn=2.3719 equaling 3.2421825×102[BK(Epo,ρ,Z=6)C0.85]/ε, [BK(Epo,ρ,Z=6)C0.85]/εequaling 1.4188×10-2is obtained; according to the relation between the experimentalδ1.5 keVof the NEA diamond withEpomax=0.85keV equaling 4.35[4]and theδ1.5 keVcalculated with Eq.(34),Epo=1.5keV andn=2.3719 equaling 2.9853632×102[BK(Epo,ρ,Z=6)C0.85]/ε, [BK(Epo,ρ,Z=6)C0.85]/εequaling 1. 457×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C0.85]/εequaling1. 43×10-2is obtained.

According to the parameters of NEA diamond withEpomax=0.85keV (n=2.3719,K(Epo,ρ,Z=6)C0.85(B/ε) =1. 43×10-2) and Eq.(34), theδat 0.8keV≤Epo≤2keV of NEA diamond withEpomax=0.85keV can be expressed as:

(35)

From the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of NEA diamond equalsK(Epo,ρ,Z=6)C0.85, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α= 87.89 ?,r=0. 064,K(Epo,ρ,Z=6)C0.85(B/ε)=1.43×10-2,Epomax=0.85keV) and Eqs.(2) and (4), theδat 2keV≤Epo≤3keV of NEA diamond withEpomax=0.85keV can be expressed as:

(36)

9 SEE from NEA diamond with Epomax=1.1keV

Seen from Fig. 5, it is known that theEpomaxof NEA diamond withEpomax=1.1keV is 1.1keV[4].R1.1keV calculated with Eq.(1) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=1.1keV) is equal to 293.9866 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=1.1keV can be expressed as:

(37)

Seen from Fig. 5, it is known that theEpomaxof NEA diamond withEpomax=1.1keV is in the range of 0.8keV≤Epomax≤2keV.Thus, from the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2, we take theK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of the NEA diamond withEpomax=1.1keV to be a constantK(Epo,ρ,Z=6)C1.1; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,r=0.064,Epomax=1.1keV), the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3 keV of the NEA diamond withEpomax=1.1keV equalsK(Epo,ρ,Z=6)C1.1and Eqs.(1), (4) and (37), theδat 0.8keV≤Epo≤2keV of the NEA diamond withEpomax=1.1keV can be expressed as follows:

(38)

Theδat 0.8keV≤Epo≤2keV of the NEA diamond reaches itsδmatEpo=1.1keV. Thus, from Eq.(38), the result thatnof Eq.(38) approximately equals 2.3849 is obtained. Therefore, the (1/α) of NEA diamond withEpomax=1.1keV calculated with Eq.(37) andn=2.3849 is equal to 123.27 ?. Based onthe relation between experimentalδ1.1 keVof the NEA diamond withEpomax=1.1keV equaling 10[4]and theδ1.1 keVcalculated with Eq.(38),Epo=1.1keV andn=2.3849 equaling 4.225143×102[BK(Epo,ρ,Z=6)C1.1]/ε, [BK(Epo,ρ,Z=6)C1.1]/εequaling 2.36678×10-2is obtained; according to the relation between the experimentalδ1.9 keVof the NEA diamond withEpomax=1.1keV equaling 9[4]and theδ1.9 keVcalculated with Eq.(38),Epo=1.9keV andn=2.3849 equaling 3.8833×102[BK(Epo,ρ,Z=6)C1.1]/ε, [BK(Epo,ρ,Z=6)C1.1]/εequaling 2.314×10-2is obtained; on the basis of the relation between the experimentalδ0.8 keVof the NEA diamond withEpomax=1.1 keV equaling 9.8[4]and theδ0.8 keVcalculated with Eq.(38),Epo=0.8keV andn=2.3849 equaling 4.07572×102[BK(Epo,ρ,Z=6)C1.1]/ε, [BK(Epo,ρ,Z=6)C1.1]/εequaling 2.404×10-2is obtained; according to the relation between the experimentalδ1.5 keVof the NEA diamond withEpomax=1.1keV equaling 9.8[4]and theδ1.5 keVcalculated with Eq.(38),Epo=1.5keV andn=2.3849 equaling 4.0994766×102[BK(Epo,ρ,Z=6)C1.1]/ε, [BK(Epo,ρ,Z=6)C1.1]/εequaling 2.39×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C1.1]/εequaling 2.3687×10-2is obtained.

According to the parameters of NEA diamond withEpomax=1.1keV (n=2.3849,K(Epo,ρ,Z=6)C1.1(B/ε) =2.3687×10-2) and Eq.(38), theδat 0.8keV≤Epo≤2keV of NEA diamond withEpomax=1.1keV can be expressed as:

(39)

From the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of NEA diamond equalsK(Epo,ρ,Z=6)C1.1, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α= 123.27 ?,r=0. 064,K(Epo,ρ,Z=6)C1.1(B/ε)=2.3687×10-2,Epomax=1.1keV) and Eqs.(2) and (4), theδat 2keV≤Epo≤3keV of NEA diamond withEpomax=1.1keV can be expressed as:

(40)

10 SEE from NEA diamond with Epomax=1.72keV

Seen from Fig. 2, it is known that theEpomaxof NEA diamond withEpomax=1.72keV is 1.72keV[4].R1.72 keVcalculated with Eq.(1) and parameters of diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=1.72keV) is equal to 533.548 ?. Therefore, from Eq.(7), the (1/α) of NEA diamond withEpomax=1.72keV can be expressed as:

(41)

Seen from Fig. 2, it is known that theEpomaxof NEA diamond withEpomax=1.72keV is in the e range of 0.8keV≤Epomax≤2keV.Thus, from the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on as a constantK(Epo,ρ,Z)C0.8-2, we take theK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of the NEA diamond withEpomax=1.72keV to be a constantK(Epo,ρ,Z=6)C1.72; and the ratio ofBtoεis independent ofEpo[22-24]. Therefore, from parameters of NEA diamond[4, 21](ρ=3.52g/cm3,Aα=12,Z=6,r=0.064,Epomax=1.72keV), the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3 keV of the NEA diamond withEpomax=1.72keV equalsK(Epo,ρ,Z=6)C1.72and Eqs.(1), (4) and (41), theδat 0.8keV≤Epo≤2keV of the NEA diamond withEpomax=1.72keV can be expressed as follows:

(42)

Theδat 0.8keV≤Epo≤2keV of the NEA diamond reaches itsδmatEpo=1.72keV. Thus, from Eq.(42), the result thatnof Eq.(42) approximately equals 2.4195 is obtained. Therefore, the (1/α) of NEA diamond withEpomax=1.72keV calculated with Eq.(41) andn=2.4195 is equal to 220.52 ?. Based onthe relation between experimentalδ1.72 keVof the NEA diamond withEpomax=1.72keV equaling 20[4]and theδ1.72 keVcalculated with Eq.(42),Epo=1.72keV andn=2.4195 equaling 6.57654×102[BK(Epo,ρ,Z=6)C1.72]/ε, [BK(Epo,ρ,Z=6)C1.72]/εequaling 3.04×10-2is obtained; according to the relation between the experimentalδ1.9 keVof the NEA diamond withEpomax=1.72keV equaling 19.9[4]and theδ1.9 keVcalculated with Eq.(42),Epo=1.9keV andn=2.4195 equaling 6.555147×102[BK(Epo,ρ,Z=6)C1.72]/ε, [BK(Epo,ρ,Z=6)C1.72]/εequaling 3.036×10-2is obtained; on the basis of the relation between the experimentalδ0.9 keVof the NEA diamond withEpomax=1.72keV equaling 16[4]and theδ0.9 keVcalculated with Eq.(42),Epo=0.9keV andn=2.4195 equaling 5.68106×102[BK(Epo,ρ,Z=6)C1.72]/ε, [BK(Epo,ρ,Z=6)C1.72]/εequaling 2.816×10-2is obtained; according to the relation between the experimentalδ1.3 keVof the NEA diamond withEpomax=1.72keV equaling 18.1[4]and theδ1.3 keVcalculated with Eq.(42),Epo=1.3keV andn=2.4195 equaling 6.3985×102[BK(Epo,ρ,Z=6)C1.72]/ε, [BK(Epo,ρ,Z=6)C1.72]/εequaling 2.829×10-2is obtained. Thus, the average value of [BK(Epo,ρ,Z=6)C1.72]/εequaling 2.93×10-2is obtained.

According to the parameters of NEA diamond withEpomax=1.72keV (n=2.4195,K(Epo,ρ,Z=6)C1.72(B/ε) =2.93×10-2) and Eq.(42), theδat 0.8keV≤Epo≤2keV of NEA diamond withEpomax=1.72keV can be expressed as:

(43)

From the assumption thatK(Epo,ρ,Z=6) at 0.8keV≤Epo≤3keV of NEA diamond equalsK(Epo,ρ,Z=6)C1.72, parameters[4, 21](ρ=3.52g/cm3,Aα=12,Z=6, 1/α= 220.52 ?,r=0. 064,K(Epo,ρ,Z=6)C1.72(B/ε)=2.93×10-2,Epomax=1.72keV) and Eqs.(2) and (4), theδat 2keV≤Epo≤3keV of NEA diamond withEpomax=1.72keV can be expressed as:

(44)

11 Formula for B

Theδat 10keV of NEA semiconductors can be expressed as[12]:

(45)

wherexis the distance from the position to the surface of semiconductor, andEpxis primary energy at a givenx.

Based on Eq.(3), the average energy loss of primary electron per unit path length dEpx/dxat 10keV can be expressed as:

(46)

For NEA semiconductors with 0.8keV≤Epomax≤2keV, theRat 10keV is much larger than the maximum escape depth of secondary electronsT, andTis approximately equal to 5/α[25]. For example, from Section.8, it is known that 1/αof NEA diamond withEpomax=0.85keV equals 87.89 ?, and thatTof NEA diamond withEpomax=0.85keV is 439.45 ?. TheRat 10 keV in NEA diamond withEpomax=0.85 keV calculated with Eq.(3) and parameters of diamond[21](ρ=3.52g/cm3,Aα=12,Z=6,Epo=10keV) is equal to 9718.79 ?. Thus, most of primary energy is dissipated outsideT, and the primary energy changes little insideT. Then, from Eq.(46), the dEpx/dxat 10keV insideTof NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately written as:

(47)

TheRat 10keV is much larger than 5/αof NEA semiconductors with 0.8keV≤Epomax≤2keV, the internal secondary electrons excited outside 5/αcan not be emitted into vacuum[25]. Thus, the definite integral [0,R] of Eq.(45) can be replaced with [0, 5/α] when primary electron at 10keV enter NEA semiconductors with 0.8keV≤Epomax≤2keV. Soδat 10keV of NEA semiconductors with 0.8keV≤Epomax≤2keV can be obtained by combining Eqs.(45) and (47):

(48)

Based on the fact that theRat 10keV is much larger than corresponding 1/αand Eqs.(3) and (6), theδat 10keV of NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately written as:

(49)

From Sections 2-10, [BK(Epo,ρ,Z)]/εat 0.5Epomax≤Epo≤10Epomaxof NEA semiconductors with 2keV≤Epomax≤5keV and that at 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤5keV can be expressed as follows:

(50)

From Sections 2-10, CNEA(Epomax,ρ,Z) of a given NEA semiconductor with 0.8keV≤Epomax≤5keV is a constant.

According to the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV can be approximately looked on as a constantK(Epo,ρ,Z)C2-5, it can be concluded that thatK(Epo,ρ,Z)of the NEA semiconductors decreases extremely slowly with increasingEpoin the range of 2keV≤Epo≤20keV. According to the fact that Eq.(48) equals Eq.(49), it is known that the [BK(Epo,ρ,Z)]/εatEpo=10keV of NEA semiconductors with 0.8keV≤Epomax≤2keV equals 0.6. Thus, from assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤2 keV approximately equals constant and conclusion thatK(Epo,ρ,Z)at 2keV≤Epo≤20keV decreases extremely slowly with increasingEpo, it can be concluded that the [BK(Epo,ρ,Z)]/εat 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤2keV approximately equal 0.6. Therefore, from Eq.(50) and the conclusion that the [BK(Epo,ρ,Z)]/εat 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV also equals 0.6[12], [BK(Epo,ρ,Z)]/εat 0.5Epomax≤Epo≤10Epomaxof NEA semiconductors with 2keV≤Epomax≤5keV and that at 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤5keV can be expressed as follows:

B=1.6667εCNEA(Epomax,ρ,Z)

(51)

Whereεcan be expressed as[26]:

(52)

whereEgandχare the width of forbidden band and the efficient electron affinity, respectively. Theχof NEA semiconductors can considered as 0, and theεof NEA diamond calculated withχ=0,Eg=5.47eV[27]and Eq.(52) is shown in Table.1, theεof NEA GaN calculated withχ=0,Eg=3.2eV[11]and Eq.(52) is also shown in Table.1.

The formula forδof NEA semiconductors which is used by some authors to analyzeBof NEA semiconductors is written as[22]:

(53)

Tab.1 Parameters of NEA diamond and GaN with 0.8keV≤Epomax≤5keV

12 Results and discussion

Theδof NEA GaN and three diamond with 2keV≤Epomax≤5keV calculated with correspondingEpoand Eqs.(10), (11), (12), (15), (16), (19), (20), (23) and (24) are shown in Figs.1-3[4, 13]. Seen from Figs. 1 and 3[4, 13], it is known that the calculatedδof NEA GaN withEpomax=3.0keV and diamond withEpomax=2.3keV agree very well with corresponding experimental ones[4, 13].Seen from Figs. 2 and 3[4], as a whole, it is known that the calculatedδof NEA diamond withEpomax=2.75keV and diamond withEpomax=2.64keV agree with corresponding experimental ones[4]. But there are some differences between some calculatedδof NEA diamond withEpomax=2.75keV and diamond withEpomax=2.64keV and corresponding experimental ones. We assume that four factors may lead to this result. First, primary electron impingement modified the surface termination and thus altered theδduring the course of measuringδ. The larger primary current was used during the measurement of theδof NEA diamond withEpomax=2.75keV and diamond withEpomax=2.64keV[4]. Second, there are larger experimental errors in the experimentalδof NEA diamond withEpomax=2.75keV and diamond withEpomax=2.64keV. Third, from the course of deducing Eqs.(15), (26), (19), (20), (23) and (24), it is known that the larger experimental errors in theEpoandδof the NEA diamond which are used to calculate [BK(Epo,ρ,Z=23)C]/εcan lead to some difference between realδand calculated ones.Fourth, there is an approximation thatK(Epo,ρ,Z=23) at 0.5Epomax≤Epo≤10Epomaxof NEA diamond with 2keV≤Epomax≤5keV ≈K(Epo,ρ,Z=23)C2-5made in the course of deducing the Eqs.(15), (26), (19), (20), (23) and (24).Thus, it can be concluded that Eqs.(10), (11) and (12) can be used to calculate theδat 1.5keV≤Epo≤30keV of NEA GaN withEpomax=3.0keV, and that (15), (16), (19), (20), (23) and (24) can be approximately used to calculate theδat 0.5Epomax≤Epo≤3keV of corresponding NEA diamond. Therefore, the method of deducing the formulas forδat 0.5Epomax≤Epo≤10Epomaxof NEA semiconductors with 2keV≤Epomax≤5keV, which has been proved to be correct in our former study[12], has been further proved to be correct.

There is only one assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV ≈K(Epo,ρ,Z)C2-5made in the course of deducing Eqs.(10), (11), (12), (15), (16), (19), (20), (23) and (24). So the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV ≈K(Epo,ρ,Z)C2-5, which has been proved to be correct in our former study[12], has been further proved to be correct.

Theδat 0.8keV≤Epo≤3keV of two NEA GaN and three NEA diamond with 0.8keV≤Epomax≤2keV calculated with correspondingEpoand Eqs.(27), (28), (31), (32), (35), (36), (39), (40), (43) and (44) are shown in Figs. 2, 4 and 5[4, 14]. Seen from Figs. 2, 4 and 5[4, 14], as a whole, it is known that the calculatedδof the two NEA GaN and three NEA diamond with 0.8keV≤Epomax≤2keV agree well with corresponding experimental ones[4, 14]. But there are some differences between the experimentalδat 2.45keV≤Epo≤2.9keV of the NEA diamond withEpomax=1.72keV[4]and corresponding calculated ones. According to the shape ofδand the fact that experimentalδof the NEA diamond withEpomax=1.72keV reachesδmat 1.72keV, theδat 2.45keV≤Epo≤2.9keV of the NEA diamond withEpomax=1.72keV should decrease with increasingEpo. But the experimentalδat 2.45keV≤Epo≤2.9keV of the NEA diamond withEpomax=1.72keV increase with increasingEpo. So we assume two factors may mainly lead to this result. First, there are larger experimental errors inδat 2.45keV≤Epo≤2.9keV of the NEA diamond withEpomax=1.72keV. Second, primary electron impingement modified the surface termination and thus altered theδat 2450eV≤Epo≤2.9keV during the course of measuringδof the NEA diamond withEpomax=1.72keV. Thus, it can be concluded that Eqs.(27), (28), (31), (32), (35), (36), (39), (40), (43) and (44) can be used to calculate theδat 0.8keV≤Epo≤3keV of corresponding NEA semiconductors with 0.8keV≤Epomax≤2keV, and that the method of deducing the formulas forδat 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤2keV is correct.

There is only one assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV≈K(Epo,ρ,Z)C0.8-2made in the course of deducing Eqs.(27), (28), (31), (32), (35), (36), (39), (40), (43) and (44). Thus, the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of the NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on asK(Epo,ρ,Z)C0.8-2is correct. Therefore, from the fact that the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV approximately equalK(Epo,ρ,Z)C2-5, it can be concluded that Eq.(51) deduced from some existed formulas and the two assumptions is correct. TheBof NEA diamond and GaN with 0.8keV≤Epomax≤5keV calculated with Eq.(51), corresponding CNEA(Epomax,ρ,Z) andεshown in Table.1 are still shown in Table.1.

Up to now, none have deduced formulas forBof NEA emitters. Some authors obtained theBof NEA semiconductorsby fitting Eq.(53) to the experimental data[13, 28], and theBof NEA GaN withEpomax=3.0keV and that of NEA GaP withEpomax=5.0keV obtained by the authors are 0.36 and 0.33, respectively. TheBof NEA GaN withEpomax=3.0keV calculated with parameters shown in Table.1 and Eq.(51) is 0.5418, and theBof NEA GaP withEpomax=5.0keV calculated with parameters (ε=6.3552eV,CNEA(Epomax,ρ,Z)=6.44×10-2) shown in Table.1of our former study[12]and Eq.(51) is 0.68226. Seen from comparison between theBof NEA GaN withEpomax=3.0keV and NEA GaP withEpomax=5.0keV obtained by the authors[13, 28]and correspondingBcalculated by us, it is known that theBcalculated by us are about 1.6667 times of correspondingBobtained by the authors[13, 28]. Seen from Eqs.(4), (6) and (51) and the courses of deducing Eqs.(4), (6) and (51), it is known that the important factor that dEpx/dxincreases with increasingx[12]or parameterK(Epo,ρ,Z) was taken into account in the course of deducing Eqs.(4), (6) and (51), and that this important factor was not taken into account in the course of deducing Eq.(53)[13, 22, 28]. According to the physical mechanism of SEE, the parameterK(Epo,ρ,Z) must be taken into account in the course of deducing formula forδ[12, 23, 29-31]. From the courses of deducing Eqs.(4), (6) and (51) and calculatingBwith Eq.(51), it is known that theBof NEA GaN withEpomax=3.0keV and NEA GaPwithEpomax=5.0keV calculated by us approximately equal correspondingBobtained by the authors if parameterK(Epo,ρ,Z) is not taken into account or taken to be 1. Thus, from above analysis, it concludes that theBof NEA semiconductors calculated with Eq.(51) deduced from Eqs.(4), (6) and some existed formulas are more reasonable than theBobtained by the authors by fitting Eq.(53) to the experimental data, and that Eq.(51) can be used to calculate theBof NEA semiconductors with 0.8keV≤Epomax≤5keV.

Up to now, none of formulas for 1/αof NEA semiconductors was deduced, and the 1/αof NEA semiconductors were not measured experimentally. The expression ofRis important for authors to obtain the 1/αof NEA semiconductors by fitting Eq.(53) to the experimental data. For example, when some authors obtained 1/αof NEA GaN withEpomax=3.0keV by using the expression ofR[R=0.01(Epo)2(μm),Epoin keV] and fitting Eq.(53) to the experimental data, the obtained 1/αof NEA GaN withEpomax=3keV is 300 ?[13]; when some authors obtained 1/αof NEA GaN withEpomax=3.0keV by using the expression ofR[R=0.027(Epo)2(μm),Epoin keV] and fitting Eq.(53) to the experimental data, the obtained 1/αof NEA GaN withEpomax=3.0keV is 820 ?[13].Finally, the authors assumed that the 1/αof NEA GaN withEpomax=3.0keV was estimated to be between 300 and 800 ?[13]. The expression ofRis also important for us to obtain the 1/αof NEA semiconductors with 0.8keV≤Epomax≤5keV. A problem arises, are Eqs.(1)，(2) and (3) suitable to diamond and GaN According to the total stopping powers calculated with ESTAR program[32]and Eqs.(1)，(2) and (3), we found that Eqs.(1)，(2) and (3) are suitable to diamond, GaN, GaP, GaAs, etc. For example, based on Eq.(3), the dEpx/dxat 10keV≤Epo≤100keV can be expressed as Eq.(47), and the dEpx/dxin diamond at 20keV calculated with Eq.(47) and corresponding parameters is equal to 0.3915 eV/?; the total stopping power (i.e.,dEpx/dx) in diamond at 20keV calculated with ESTAR program is equal to 11.69MeV·cm2/g,ρof diamond is equal to 3.52g/cm3. Thus, the dEpx/dxin diamond at 20keV calculated with ESTAR program is equal to (11.69MeV·cm2/g) (3.52g/cm3), that is, the dEpx/dxin diamond at 20keV calculated with ESTAR program is equal to 0.4115eV/?.Therefore,the dEpx/dxin diamond at 20keV calculated with Eq.(47) approximately equals that calculated with ESTAR program. We found that dEpx/dxin diamond at 10keV≤Epo≤100keV calculated with Eq.(47) approximately equal corresponding those calculated with ESTAR program by similar method. Hence, it can be concluded that Eq.(3) is suitable to diamond. Therefore, from the fact that Eqs.(1)，(2) and (3) are suitable to diamond, GaN, GaP, GaAs, etc, it can be concluded that the method presented here of calculating the 1/αof NEA semiconductors with 0.8keV≤Epomax≤5keV is correct, and that the 1/αof NEA semiconductors with 0.8keV≤Epomax≤5keV are correct.

Electron beam impingement can modify the surface termination of NEA diamond and thus alter theδ,Band 1/αof NEA diamond[4]. The NEA diamond withEpomax=1.1keV is boron (B)-doped and hydrogen (H) terminated NEA diamond[4]; the NEA diamond withEpomax=2.64keV is B-doped and H terminated NEA diamond after 880s of electron beam impingement atJ=4.9×10-4A/cm2[4]. In other words, the NEA diamond withEpomax=2.64keV studied in this study is the NEA diamond withEpomax=1.1keV studied in this study after 880s of electron beam impingement atJ=4.9×10-4A/cm2[4]. The NEA diamond withEpomax=2.75keV is the NEA diamond withEpomax=2.64keV after further electron beam impingement[4]. Considering theBand 1/αof NEA diamond shown in Table.1, it is known that the 1/αof NEA diamond withEpomax=2.75keV is larger than that of NEA diamond withEpomax=2.64keV which is also larger than that of NEA diamond withEpomax=1.1keV, and that theBof NEA diamond withEpomax=2.75keV is less than that of NEA diamond withEpomax=2.64keV which is also less than that of NEA diamond withEpomax=1.1keV. Thus, according to the relationships among the NEA diamond withEpomax=2.75keV, NEA diamond withEpomax=2.64keV and NEA diamond withEpomax=1.1keV, it can be concluded that the electron beam impingement can increases the 1/αof B-doped and H terminated NEA diamond by modifying the surface termination, and that the electron beam impingement can decreases theBof B-doped and H terminated NEA diamond by modifying the surface termination. If we have more experimentalδmandEpomaxand the information of sample preparations of NEA emitters, we can obtain more quantitative influences of sample preparations onBand 1/αof NEA semiconductors by above method. Thus, from the fact that sample preparations of a given NEA semiconductor decide theδat givenEpo,δm,Band 1/αand the fact that theBand 1/αof a given kind of semiconductor almost decide the value ofδmand theδat givenEpo, it concludes that the theoretical research ofBand 1/αhelp to research quantitative influences of sample preparation on SEE from NEA semiconductors with 0.8keV≤Epomax≤5keV and produce desirable NEA emitters such as NEA diamond.

13 Conclusions

According to the characteristics of SEE from NEA semiconductors with 0.8keV≤Epomax≤5keV,R, existing universal formulas forδof NEA semiconductors[12]and experimental data[4, 13, 14], special formulas forδat 0.5Epomax≤Epo≤10Epomaxof NEA diamond and GaN with 2keV≤Epomax≤5keV andδat 0.8keV≤Epo≤3keV of NEA diamond and GaN with 0.8keV≤Epomax≤2keV were deduced and experimentally proved, respectively.There is only one assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤2keV≈K(Epo,ρ,Z)C0.8-2made in the course of deducing Eqs.(27), (28), (31), (32), (35), (36), (39), (40), (43) and (44). Thus, the assumption thatK(Epo,ρ,Z) at 0.8keV≤Epo≤3keV of NEA semiconductors with 0.8keV≤Epomax≤2keV can be approximately looked on asK(Epo,ρ,Z)C0.8-2is correct. Therefore, from the fact that the assumption thatK(Epo,ρ,Z) at 0.5Epomax≤Epo≤10Epomaxof the NEA semiconductors with 2keV≤Epomax≤5keV approximately equalK(Epo,ρ,Z)C2-5, it can be concluded that Eq.(51) forBof NEA semiconductors with 0.8keV≤Epomax≤5keV deduced from some existed formulas and the two assumptions is correct.

According to the fact that Eqs.(1)-(3) are suitable to diamond, GaN, GaP, GaAs, the courses of calculating 1/αin Sections 2-10 and the comparison between the 1/αcalculated in Sections 2-10 and the 1/αdetermined by other authors[13], it can be concluded that the method presented here of calculating the 1/αof NEA semiconductors with 0.8keV≤Epomax≤5keV is correct, and that the obtained 1/αof NEA diamond and GaN with 0.8keV≤Epomax≤5keV are correct. From the conclusion that the theoretical research ofBand 1/αin this study are correct, the relationships amongδm,δ,Band 1/αand the fact that sample preparations of a given NEA emitter decide theBand 1/α, it concludes that the theoretical research ofBand 1/αin this study help to research quantitative influences of different sample preparations on SEE from NEA semiconductors and produce desirable NEA emitters such as NEA diamond.