CC BY
7
Condensed matter physics
DOI: 10.18721/JPM.13301 UDC 621.38+539.1
CONTRIBUTION OF INTERNAL IONIZATION PROCESSES IN SEMICONDUCTORS TO RADIATIVE LOSSES OF RELATIVISTIC ELECTRONS
A.E. Vasiliev1, V.V. Kozlovski1, S.N. Kolgatin2
1 Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation;
2 Bonch-Bruevich St. Petersburg State University of Telecommunications, St. Petersburg, Russian Federation
The study presents analysis of mass radiative energy losses (RL) incurred by relativistic electrons in different materials commonly used in semiconductor electronics. We have specifically focused on accounting for the processes of 'internal' ionization, resulting in the production of electron-hole pairs in semiconductors and dielectrics. We have established that accounting for these processes is the only method offering consistent explanations on the values of mass RLs observed experimentally. The analysis performed should allow to make more detailed predictions for the performance of semiconductor devices in real conditions, particularly, in space.
Keywords: relativistic electron, ionization potential, radiative energy losses, silicon, germanium, graphene
Citation: Vasiliev A.E., Kozlovski V.V., Kolgatin S.N., Contribution of internal ionization processes in semiconductors to radiative losses of relativistic electrons, St. Petersburg Polytechnical State University Journal. Physics and Mathematics. 13 (3) (2020) 7-14. DOI: 10.18721/JPM.13301
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons.org/ licenses/by-nc/4.0/)
ВКЛАД ПРОЦЕССОВ ВНУТРЕННЕЙ ИОНИЗАЦИИ ПОЛУПРОВОДНИКОВ В ТОРМОЗНЫЕ ПОТЕРИ ЭНЕРГИИ РЕЛЯТИВИСТСКИХ ЭЛЕКТРОНОВ
А.Э. Васильев1, В.В. Козловский1, С.Н. Колгатин2
1 Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация;
2 Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича, Санкт-Петербург, Российская Федерация
Выполнен анализ массовых тормозных потерь энергии (ТПЭ) релятивистских электронов в различных материалах, используемых в полупроводниковой электронике. Особое внимание уделено учету процессов «внутренней» ионизации, приводящей к образованию электронно-дырочных пар в полупроводниках и диэлектриках. Показано, что только при таком учете удается непротиворечиво объяснить экспериментально наблюдаемые значения массовых ТПЭ. Проведенный в работе анализ позволит выполнять более детальное прогнозирование работоспособности полупроводниковых приборов в реальных, в частности космических, условиях.
Ключевые слова: релятивистский электрон, потенциал ионизации, тормозные потери энергии, кремний, германий, графен
Ссылка при цитировании: Васильев А.Э., Козловский В.В., Колгатин С.Н. Вклад процессов внутренней ионизации полупроводников в тормозные потери энергии релятивистских элек-
тронов // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2020. Т. 13. № 3. С. 7-14. DOI: 10.18721/JPM.13301
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https://creative-commons.org/licenses/by-nc/4.0/)
Introduction
While the effects of electron irradiation on the properties of semiconductor structures and devices have been considered in numerous papers and books [1 — 4], many aspects of this problem are yet to be fully understood. Most studies tend to focus on the role of elastic processes and the effect of emerging radiation defects on the properties of materials and devices [4 — 8]. The contribution from inelastic energy losses of bombarding particles is discussed to a far lesser extent. However, it is the inelastic processes that determine the resistance to electron radiation for a number of semiconductor devices, e.g., metal-oxide-semiconductor (MOS) structures and field-effect transistors [9].
The goal of our study is to investigate the ionization losses and the absorbed energy of relativ-istic electrons in different materials used in semiconductor electronics. In particular, we concentrated on the processes of 'internal' ionization leading to production of electron-hole pairs in semiconductors and dielectrics. Relativistic electrons of 0.9 MeV (V = 0.94c) were used as irradiating particles. The particles and the energy were chosen so that the computational data could be verified experimentally with the RTE-1V electron accelerator available at Peter the Great St. Petersburg Polytechnic University.
Estimation of radiative energy losses of relativistic electrons within the Born approximation of scattering theory
In general, calculating the absorbed energy is a complex problem that can be best solved by numerical methods. We confine ourselves to considering the situation when the thickness of the irradiated sample is much lower than the particle range, which is the case in most applied problems.
The absorbed dose De depends on linear radiative losses (RL) of the bombarding electrons (dE/dx) in the medium:
D = (1/p)(dE/dX)F .
(1)
Here p is the density of the medium, Fe is the exposure dose, often referred to as fluence.
The quantity (1/p)- (dE/dx) which is called the reduced (or mass) RL, is more common in practice. For convenience, Eq. (1) can be transformed by introducing the units widely used for the quantities included in this formula:
De = 1.610-10(l/p)(dE/dX)Fe; (2)
De is given here in grays (Gy), mass RL (1/p)x x (dE/dx) in MeV-(cm2/g), Fe in cm-2.
Eq. (2) allows calculating the absorbed dose at a known particle fluence. The inverse formula for estimation of the fluence required to obtain a known absorbed dose takes the following form:
F =■
1
1.6-10-10 (1/p)-(dE/dx)'
(3)
The stopping power of MeV electrons is mainly due to ionization and excitation of bound electrons in target atoms (ionizing losses). Therefore, the notions of radiative and ionizing energy losses are virtually identical in this case. Ionizing energy losses (IEL) of relativistic electrons due to excitation and ionization of target electrons are described by the Bethe formula obtained within the Born approximation of scattering theory [10]:
2 pNatZe4
ln
meV2 E 2 12(1 -ß2) ~
+ 1 -ß2 +
, mV2
ln2 (2(1 -ß2 )1/2 -1 +ß2) 2\1/2-i2 Л
+
2 , [1-(1-ß2 )12]
(4)
where E is the kinetic energy of the relativistic electron, Vis the velocity of the incident electron,
x
ß = V/c is the relativistic factor, I is the mean ionization potential of the target atoms.
IEL linearly depend on the number of electrons per unit of the target volume (electron density), N.. Electron density, in turn, is known to be proportional to the density of the medium:
N = ZN t = Z ■ p ■ N0/A.
e at ' 0
(5)
dE dx
Z
8nNABOr ER 2r0 2
(
ln
meß2c2 E
\
(10)
A-ß21 212(1 -ß2)
Or, substituting the universal constants, we arrive to:
Here N0 is the Avogadro constant; A is the atomic mass of the medium.
The first (logarithmic) term in curly brackets in the Bethe formula (4) exceeds the remaining terms by an order of magnitude in the given examples. For this reason, Eq. (4) can be simplified by omitting all terms except the first one.
'dE
dx
2nNatZe
4 f
mV2
ln-
meV2 E 2 12(1 -ß2)
A
1 ( dE
p V dx
( MeV - cm2/g ):
= 0.154 -
A-ß2
2
ln
0.511-106-ß2 E 212 (1 -ß2 )
(11)
(6)
It is often assumed that normalized linear IEL reduced to electron density in the target (or normalized mass RL reduced to mass-to-charge ratio of the target nucleus),
Let us express the squared initial velocity of the incident electron in terms of the relativistic factor
dE dx
2nNatZe
4 f
mß c2
ln
meß2c2E ^ 2 12(1 -ß2 )
Let us rewrite the factor in front of the logarithm in expression (7), introducing the Rydberg energy Er and the Bohr radius r0 widely used in atomic physics:
dE \ _ 8nNatZER2r2
v dx )wn meß2c2
ln
meß2c2 E 212(1 -ß2 )
m e
Er =--e 0 , = 13.6 eV,
2(4ns0) h2
4nsn h
2
= 0.53-10-8 cm.
me
Now Eq. (8) can be used for linear IEL to obtain the formula for mass RL (1/p) • (dE/dx), given that the density of the medium p = A • Na/N0:
dE dx
_1 fln meß2c2e >
ß2 V 212(1 -ß2)
(12)
= K ( Z ),
. (7)
is a quantity independent of the material of the stopping target, equal to 18/P2 [11]. This implies that the contribution from variation of the mean ionization potential under the logarithm in Eq. (11) is small. Making this assumption, we can use Eq. (11) to easily calculate mass RL in any medium based on the experimentally found RL, for example, in aluminum [12]:
(8)
(
1 dE
/Al
(9)
(13)
We believe that neglecting the contribution from the ionization potential of the target atoms and using Eq. (13) is ill-suited for our problems. For this reason, Eq. (11) was used to calculate RL in some materials common for semiconductor electronics. Semiconductors with different band gaps, and metals with different ohmic and
x
mec
X
X
X
r0 =
Eq. (15) was used to calculate the stopping powers of silicon oxide (dielectric) and silicon carbide (wide-bandgap semiconductor). Both values of 1/p(dE/dx) coincided and were equal to 1.61 MeV(cm2/g).
Fig. 1 shows the curve for mass RL normalized to mass-to-charge ratio of the target nucleus K(Z), as a function of the nuclear charge; the dependence was obtained by Eq. (12). This curve can be extrapolated by the dependence
KV-X*} (16)
As follows from Fig. 1, substituting the curve with a straight line K(Z) = 18/P2 is not entirely acceptable for light and heavy targets.
According to the Bethe formula, targets with close values of Z should also have close values of mass RL. For example, it can be seen from Table 1 that the elements with Z = 13 (aluminum) and Z = 14 (silicon) have virtually the same calculated values of mass RL (1.53 and 1.56 MeVx x(cm2/g), respectively). However, it was experimentally confirmed in Ref. [14] that RL in silicon are higher than in aluminum by almost 1.5 times (1.5 and 2.1 MeV(cm2/g), respectively). Possible explanations for this difference may lie in the mechanism of internal ionization in semiconductors.
T a b l e 1 Mean ionization potentials, mass RL, and coefficients for converting absorbed dose to fluence (and vice versa) for irradiation of different materials used in modern semiconductor electronics with 0.9 MeV electrons
Target material I (1/p)-(dE/dx) F/D
eV MeV(cm2/g) 1/Gycm2
Graphene 69 1.72 3.6109
Aluminum 150 1.53 4.1109
Silicon 161 1.56 4.0109
Germanium 288 1.29 4.8109
Gold 711 1.04 6.0109
rectifying contacts were selected. The mean ion-ization potential I and mass RL of electrons were approximated for these materials. The value of I was taken equal to Ref. [13]:
I = 11.5Z (for Z < 15), (14)
I = 9.0Z (for large Z).
The data obtained are given in Table 1. The table also lists the coefficients for calculating the absorbed dose at a known fluence (by Eq. (2)) and calculating the fluence at a known absorbed dose (by Eq. (3)).
As evident from Table 1, mass RL and conversion factors between the exposure and absorbed doses can differ by 1.7 times for most materials (with the atomic number Z ranging from 6 to 79).
The Bragg rule was used for the case when the stopping medium was a chemical compound consisting of several elements [13]. According to this rule, the stopping power of a complex substance is equal to the weighted sum of stopping powers of the constituent elements:
1 dE q p dx p1
(15)
where ro1 and ro2 are the relative proportions of elements in the compound (wt. %).
Kl
20191817; 1615; 14-
¡sl-e 1
0 10 20 io 40 50 60 70 SO
10 20 30 40 50 60 70 80 90
Z
Fig. 1. Mass RL normalized to mass-to-charge ratio of target nucleus K(Z) as a function of nuclear charge; obtained by Eq. (12).
The inset shows mass RL as a function of the nuclear charge calculated by Eq. (11) within the Born approximation of scattering theory. The black dots indicate mass RL values obtained taking into account the internal ionization for graphene, silicon, and germanium
Contribution from internal ionization of semiconductors to radiative energy losses of relativistic electrons
The concept of internal ionization is introduced for condensed matter. Internal ionization in semiconductors and dielectrics corresponds to the transition of valence electrons to the conduction band (band-to-band transition). Klein [15] suggested an equation relating the energy for production of an electron-hole pair E. and the band gap E (in eV):
E = 2.67E + 0.87,
(17)
establishing that the internal ionization energy is approximately three times more than the band gap.
E . is higher than Eg because the energy of rel-ativistic electrons is spent not only for ionization but also for generation of excited states in a solid, i.e., plasmons and phonons. Table 2 gives the energies E and Eg for the main materials used in modern semiconductor electronics (silicon, germanium) and graphene.
Since the average ionization potential, which is equal to ~9Z for most elements, is significantly (by orders of magnitude) higher than the energy for the production of electron-hole pairs in semiconductors, the main result of electron stopping is a sharp increase in the concentration of charge carriers. Mass RL are estimated by substituting into Eq. (10) the energies for production of electron-hole pairs for the materials
g
T a b l e 2
Band gap, energy for production of electron-hole pairs, mass RL and pair production rate by single relativistic electron for three semiconductor materials
Target material E E. (1/pHdE/dx) N /F
eV MeV(cm2/g) cm 1
Graphene 5.2 18.7 1.94 1.6105
Silicon 1.12 3.6 2.23 1.4106
Germanium 0.67 2.9 1.99 3.7-106
listed in Table 2. The results obtained are given in column 3 of Table 2. Comparing the data in Table 1 and Table 2, we can conclude that taking into account internal ionization changes (that is, increases) the capacities for RL: for example, by almost 50 % for silicon and germanium (up to 2.23 MeV • (cm2/g)).With this factor taken into account, the calculated values of mass RL (2.23 MeV • (cm2/g)) are much closer to the experimental ones (2.21 MeV • (cm2/g)) [14]. The black dots in the inset at Fig. 1 correspond to the values of mass RL normalized to mass-to-charge ratio of the target nucleus, accounting for internal ionization for graphene, silicon, and germanium. Let us estimate the concentration of electron-hole pairs produced by a single rel-ativistic electron (Ne_JF), dividing linear RL by pair production energy. This data is given in column 4. For example, this value for silicon is 1.4 • 106 cm-1. Let us estimate the production rate of electron-hole pairs for the real electron accelerator running at Peter the Great St. Petersburg Polytechnic University. Irradiation with electrons is performed using an RTE-1V pulse accelerator. Pulse frequency is 450 Hz, pulse duration 370 ps, duty cycle 1/6. A beam with a current of 1 mA and a cross-sectional diameter of 0.9 cm scans over an area of 2 * 40 cm. The mean current density of the beam during irradiation with electrons is taken to be 12.5 pA- cm-2; however, the current density in the pulse is much higher, reaching 6 mA • cm-2. The electron flux density in the pulse at such currents is 3.6 • 1016 cm-2s-1, and the total production rate
for electron-hole pairs upon electron irradiation reaches a huge value (1.4 • 106 • 3.6 • 1016) = = 5 • 1022 cm-3s-1. An additional charge is generated upon irradiation of MOS structures and field-effect transistors at the insulator-semiconductor interface and in the bulk of the insulator due to production of electron-hole pairs, resulting in a change in the main characteristic, which is the threshold voltage of the device [9].
Summary
The results obtained in the course of our investigation led us to the following conclusions:
1. Accounting for internal ionization of semiconductors due to production of electron-hole pairs changes (increases) the stopping powers of relativistic electrons, for example, by almost 50% for silicon and germanium.
2. This in turn offers a consistent explanation for the values of mass RL observed experimentally.
3. The analysis carried out in the study should allow making more effective and more detailed predictions for the performance of semiconductor devices in real conditions, particularly, in space.
Acknowledgments
This study was supported by the Academic Excellence Project 5-100 proposed by Peter the Great St. Petersburg Polytechnic University.
We are grateful to Prof. Vadim Ivanov (Peter the Great St. Petersburg Polytechnic University) for the insights provided.
REFERENCES
1. Lebedev A.A., Materials research foundations, Vol. 6. Radiation Effects in Silicon Carbide, Material Research Forum LLC, Millersville, 2017.
2. Claeys C., Simoen E., Radiation effects in advanced semiconductor materials and devices, Springer-Verlag, Berlin, 2002.
3. Kozlovskiy V.V., Vasilyev A.E., Emtsev
V.V., et al., The Frenkel pairs formation in the sil-
icon under high energy electron and proton irradi-
ation, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. (2 (122)) (2011)
13—21 (in Russian).
4. Van Lint V.A., Flanagan T.M., Leadon R.E., et al., Mechanisms of radiation effects in electronic materials, John Wiley & Sons Inc., New York, 1980.
5. Lutz J., Schlangenotto H., Scheuermann U., De Doncker R., Semiconductor power devices physics, characteristics, reliability, Springer-Verlag, Berlin, Heidelberg, 2011.
6. Iwamoto N., Svensson B.G., Semiconductors and semimetals, Vol. 91. Part of the volume:
Defects in semiconductors, Chap. Ten — Point defects in silicon carbide, Elsevier (2015) 369—407.
7. Holmes-Siedle A., Adams L., Handbook of radiation effects, Oxford University, Oxford, 1993.
8. Lehmann C., Interaction of radiation with solids and elementary defect production, North-Holland, Amsterdam, 1977.
9. Lebedev A.A., Kozlovski V.V., Levinshtein M.E., et al., Effect of high energy (15 MeV) proton irradiation on vertical power 4H-SiC MOS-FETs, Semiconductor Science and Technology. 34 (4)(2019) 045004.
10. Bethe H.A., Ashkin J., Experimental nuclear physics, Vol. 1. Ed. by E. Segre, Wiley, New York, 1953.
11. Marion I.B., Young F.C., Nuclear reaction analysis: Graphs and tables, North Holland Pub-
lishing Co., Amsterdam, 1968.
12. Corbett J.W., Bourgoin J.C., Point defects in solids, Vol. 2. Semiconductors and molecular crystals, Defect creation in semiconductors, Ed. by J.H. Crawford and L.M. Slifkin. Springer, Boston, 1975.
13. Properties of advanced semiconductor materials: GaN, AlN, InN, BN, SiC, SiGe, Ed. by M.E. Levinshtein, S.L. Rumyantsev, M.S. Shur, John Wiley & Sons, New York, 2001.
14. Combasson J.L., Farmery B.W., McCulloch D., et al., Ion ranges in aluminum and silicon, Radiation Effects. 36 (3-4) (1978) 149-156.
15. Klein C.P., Radiation ionization energies in semiconductors - Speculations about the role of plasmons, Proceedings of the 8th International Conference on the Physics of Semiconductors, Kyoto (Japan) (1966) 307-311.
Received 17.06.2020, accepted 05.08.2020.
THE AUTHORS
VASILIEV Alexander E.
Peter the Great St. Petersburg Polytechnic University
29 Politechnicheskaya St., St. Petersburg, 195251, Russian Federation
vasiliev_ae@spbstu.ru
KOZLOVSKI Vitaly V.
Peter the Great St. Petersburg Polytechnic University
29 Politechnicheskaya St., St. Petersburg, 195251, Russian Federation
vkozlovski@phmf.spbstu.ru
KOLGATIN Sergey N.
Bonch-Bruevich St. Petersburg State University of Telecommunications 61 Moika Emb., St. Petersburg, 191186, Russian Federation kolgatins@gmail.com
СПИСОК ЛИТЕРАТУРЫ
1. Lebedev A.A. Materials research foundations. Vol. 6. Radiation effects in silicon carbide. Millers-ville: Material Research Forum LLC, 2017. 178 p.
2. Claeys C., Simoen E. Radiation effects in advanced semiconductor materials and devices. Berlin: Springer-Verlag, 2002. 401 p.
3. Козловский В.В., Васильев А.Э., Емцев В.В., Оганесян Г.А., Колгатин С.Н. Образо-
вание пар Френкеля в кремнии под действием электронов и протонов высоких энергий // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2011. № 2 (122). С. 13-21.
4. Van Lint V.A., Flanagan T.M., Leadon R.E., Naber J.A., Rogers V.C. Mechanisms of radiation effects in electronic materials. New York:
John Wiley & Sons Inc., 1980. 359 p.
5. Lutz J., Schlangenotto H., Scheuermann U., de Doncker R. Semiconductor power devices physics. Characteristics, reliability. Berlin, Heidelberg: Springer-Verlag, 2011. 536 p.
6. Iwamoto N., Svensson B.G. Semiconductors and semimetals. Vol. 91. Part of the volume: Defects in Semiconductors. Chapter Ten — Point Defects in Silicon Carbide. Elsevier, 2015. Pp. 369-407.
7. Holmes-Siedle A., Adams L. Handbook of radiation effects. Oxford: Oxford University, 1993. 479 p.
8. Lehmann C. Interaction of radiation with solids and elementary defect production. Amsterdam: North-Holland, 1977. 341 p.
9. Lebedev A.A., Kozlovski V.V., Levinshtein M.E., Ivanov P.A., Strel'chuk A.M., Zubov A.V., Fursin L. Effect of high energy (15 MeV) proton irradiation on vertical power 4H-SiC MOSFETs // Semiconductor Science and Technology. 2019. Vol. 34. No. 4. P. 045004.
10. Bethe H.A., Ashkin J. Experimental nucle-
ar physics. Vol. 1. Ed. by E. Segre. New York: Wiley, 1953. 662 p.
11. Marion I.B., Young F.C. Nuclear reaction analysis: graphs and tables. Amsterdam: North Holland Publishing Co., 1968. 169 p.
12. Corbett J.W., Bourgoin J.C. Point defects in solids. Vol. 2. Semiconductors and molecular crystals. Defect creation in semiconductors. Ed. by J.H. Crawford and L.M. Slifkin. Boston: Springer, 1975.166 p.
13. Properties of advanced semiconductor materials: GaN, AlN, InN, BN, SiC, SiGe. Ed. by M.E. Levinshtein, S.L. Rumyantsev, M.S. Shur. New York: John Wiley & Sons, 2001. 216 p.
14. Combasson J.L., Farmery B.W., McCulloch D., Nielsen G.W., Thompson M.W. Ion ranges in aluminum and silicon // Radiation Effects. 1978. Vol. 36. No. 3-4. Pp. 149-156.
15. Klein C.P. Radiation ionization energies in semiconductors - Speculations about the role of plasmons // Proceedings of the 8th International Conference on the Physics of Semiconductors. Kyoto (Japan). 1966. Pp. 307-311.
Статья поступила в редакцию 17.06.2020, принята к публикации 05.08.2020.
СВЕДЕНИЯ ОБ АВТОРАХ
ВАСИЛЬЕВ Александр Электронович — кандидат физико-математических наук, доцент кафедры экспериментальной физики Санкт-Петербургского политехнического университета Петра Великого, Санкт-Петербург, Российская Федерация.
195251, Российская Федерация, г. Санкт-Петербург, Политехническая ул., 29 vasiliev_ae@spbstu.ru
КОЗЛОВСКИЙ Виталий Васильевич — доктор физико-математических наук, профессор кафедры экспериментальной физики Санкт-Петербургского политехнического университета Петра Великого, Санкт-Петербург, Российская Федерация.
195251, Российская Федерация, г. Санкт-Петербург, Политехническая ул., 29 vkozlovski@phmf.spbstu.ru
КОЛГАТИН Сергей Николаевич — доктор технических наук, заведующий кафедрой физики Санкт-Петербургского государственного университета телекоммуникаций им. проф. М.А. Бонч-Бруевича, Санкт-Петербург, Российская Федерация.
191186, Российская Федерация, г. Санкт-Петербург, наб. р. Мойки, 61 kolgatins@gmail.com
© Санкт-Петербургский политехнический университет Петра Великого, 2020
