Electron–Hole Pair Creation Energy and Fano Factor Temperature Dependence in Silicon

ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 584 (2008) 436–439 www.elsevier.com/locate/nima Letter to the Editor Electron–hole pair creation energy and Fano factor temperature dependence in silicon

M.N. MazziottaÃ

Istituto Nazionale di Fisica Nucleare Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy

Received 14 September 2007; received in revised form 22 October 2007; accepted 31 October 2007 Available online 4 November 2007

Abstract

The energy to create an electron–hole pair and the Fano factor in silicon have been evaluated as a function of the temperature by means of a full Monte Carlo simulation. r 2007 Elsevier B.V. All rights reserved.

PACS: 07.05.Fb; 29.40.Wk

Keywords: Silicon detector; Electron–hole pair creation energy; Fano factor

Recently a measurement of the energy to create an and holes lose part of their kinetic energy by phonon electron–hole (e–h) pair and its temperature dependence in scattering and another part by impact ionizations, produ- the range between 80 and 270 K has been performed using cing secondary e–h pairs [3]. an Si PIN diode with 5:9 keV X-rays [1]. A value of 3:73 In this Monte Carlo the generation of secondary e–h 0:09 eV with a gradient of 0:0131 0:0004% K1 was pairs is simulated following the prescriptions of [4,5], where found. A Fano factor of 0:118 0:004 in the temperature a simple band-structure model is assumed and the random- range between 110 and 235 K was also found. The mean k approximation to the transition rate for impact ioniza- energy for the generation of one pair was also measured tion is used for both electrons and holes. The ratio of with synchrotron radiation with energies between 300 and phonon emission with respect to the impact ionization is 1400 eV at room temperature and at 140 K [2]. Experi- the same for both electrons and holes, and depends on the mental results show that the energy to create an e–h pair in energy gap, on the phonon energy and on parameter that is silicon depends both on the photon energy and on the assumed independent on the particle energy and invariant temperature. for electrons and holes, and for the temperature. The In this paper the energy WðE; TÞ to create an e–h pair relaxation process following the photo-absorption in the and the Fano factor FðE; TÞ in silicon as a function of the silicon shells produces electrons and vacancies in the shells photon energy and their temperature dependence are according to the relaxation trees and probabilities that evaluated by means of a full Monte Carlo simulation. have been taken from Refs. [6,7]. The charge carriers are produced both in the primary The temperature dependence of W and F are evaluated collision (e.g. when a photon is absorbed) and by the by assuming a temperature dependence of the energy gap in subsequent energy losses of the secondary e–h pairs. The silicon EgðTÞ given by the Varshni equation [8]: absorption of photons and the subsequent relaxation of the aT 2 excited atoms yield the photo-electrons, the Auger and E ðTÞ¼E ð0Þ (1) Coster–Kronig primary electrons and the corresponding g g T þ b primary holes in the valence band. All primary electrons where Egð0Þ¼1:17 eV is the energy gap at the temperature 4 ÃTel.: +39 0805443163; fax +39 0805442470. T ¼ 0K, a ¼ 4:73 10 eV=K and b ¼ 636 K. The para- E-mail address: [email protected] meters of Eq. (1) are taken from the Ioffe Institute

0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.10.043 ARTICLE IN PRESS M.N. Mazziotta / Nuclear Instruments and Methods in Physics Research A 584 (2008) 436–439 437 database [9] and they are slightly different from the results 3.74 published in Ref. [10]. Another expression for the dependence of the energy gap with temperature can be 3.72 found in Ref. [11]. The energy of the optical phonon E0 is set to a constant value of 0.063 eV in the calculation. 3.7 Fig. 1 shows the energy to create an e–h pair as a function of the temperature for incident photons of 5.9 keV 3.68 energy. The pair creation energy W is given by 3.66 E WðE; TÞ¼ (2) nðE; TÞ 3.64 where E is the photon energy and nðE; TÞ is the number of 3.62 total pairs, that takes also into account the primary e–h Pair Creation energy (eV) pairs (e.g. the photo-electron, the Auger electrons and their 3.6 vacancies) [3]. The W value is evaluated as the average value from a sample of 10,000 simulated events for each 3.58 temperature value from 10 to 460 K. The energy to create 3.56 an e–h pair slightly decreases with the temperature, from 0 50 100 150 200 250 300 350 400 450 about 3.73 eV at 10 K to about 3.57 eV at 460 K. The e–h Temperature (K) pair creation energy gradient is of order of 104 eV=K, even though the energy to create a pair is not linear with the Fig. 2. Electron–hole pair creation energy (average value) as a function of temperature. The Fano factor for the pair number the temperature at different photon energies. Full circles: E ¼ 50 eV; full triangles: E ¼ 125 eV; full squares: E ¼ 200 eV; open circles: E ¼ 500 eV. distribution is almost constant with small ﬂuctuations The dashed lines are drawn as a guide for the eye. around the value of 0.117. Fig. 2 shows the energy to create an e–h pair as a function of the temperature in the range from 50 to 400 K due to the phonon temperature dependence is negligible. In evaluated at four different photon energies in the region order to evaluate this effect, the energy to create an e–h around the Si L-shells, i.e. 50, 125, 200 and 500 eV. The e–h pair is evaluated assuming a linear dependence of the pair creation energy gradient is still of order of 104 eV=K phonon energy from the temperature, i.e. E0ðeVÞ¼ for these photon energy values. 0:063 þ aðTðKÞ300Þ. For the slope a the values of Alig et al. [4] inferred that the temperature dependence 105 and 104 eV=K have been simulated, namely 1 of the phonon energy is weak, so the contribution to the order of magnitude less than the energy gap temperature energy to create an e–h pair in silicon with the temperature gradient and of the same order of the energy gap temperature gradient. The results for incident photons of 3.8 5.9 keV energy are shown in Fig. 3. For positive values of slope a the e–h pair creation energy gradient decreases, and for large positive values it becomes positive. On the other 3.75 hand, for negative values of a the e–h pair creation energy gradient increases, and for large negative slope values it becomes too large with respect to the experimental results. 3.7 Fig. 4 shows the mean energy to produce a pair as a function of the incident photon energy, for two different 3.65 temperature values, i.e. 140 and 300 K (the energy of the optical phonon is set to a constant value of 0.063 eV). The energy to create an e–h pair approaches a constant value 3.6 for incident photon energies above the Si K-shell energy.

Pair Creation energy (eV) The constant value is about 3.70 eV at 140 K, while decreases to about 3.65 eV at 300 K. These results are in 3.55 agreement with the one shown in Refs. [2,12]. The dependence of W on the incident photon energy exhibits a discontinuity in the photon energy region around the Si 3.5 L-shells (e.g. around 100 eV) (see Ref. [3] for more details). 0 50 100 150 200 250 300 350 400 450 A maximum value of about 3.85 (3.80) eV is found at Temperature (K) incident photon energy around 4.5 eV, and a minimum Fig. 1. Electron–hole pair creation energy in silicon as a function of the value of about 3.6 (3.5) eV is found at incident photon temperature for 5.9 keV incident photons. energy around 6 eV for 140 (300) K. Finally, a linear ARTICLE IN PRESS 438 M.N. Mazziotta / Nuclear Instruments and Methods in Physics Research A 584 (2008) 436–439

3.9 As already discussed in Ref. [3], in the low energy region the energy band structure of silicon could affect the value 3.85 of the energy to create an e–h pair. In particular, the mechanism of absorption of photons in the valence band 3.8 and consequently the production of the primary e–h pair could affect the results at photon energies below 100 eV, i.e. below the L shells. In this Monte Carlo a uniform density 3.75 of states in the valence band is assumed. On the other hand, the number of secondary e–h pairs can be evaluated 3.7 in two extreme cases, i.e. when the total energy is transferred to the photo-electron (case 1), and when for 3.65 photon with energy below the valence energy value, i.e. E E , the total energy is equally shared between the

Pair Creation energy (eV) php V 3.6 photo-electron and the hole or, for higher photon energies, the maximum energy EV ¼ 12 eV is transferred to the hole 3.55 and the photo-electron energy is Epe ¼ Eph Eg EV (case 2). The results are shown in Fig. 5, with the optical 3.5 phonon energy set to a constant value of 0.063 eV (only 0 50 100 150 200 250 300 350 400 450 photon energies below 35 eV are shown). In the case 2 the Temperature (K) maximum value is found around 5.5 eV, against 3.5 eV in the case 1. The case 2 is closed to the experimental results. Fig. 3. Electron–hole pair creation energy in silicon as a function of the temperature for 5.9 keV incident photons. Solid line: a ¼ 0; open circles: The experimental results in the photon energy region a ¼104 eV=K; open squares: a ¼105 eV=K; full squares: below tens of eV could be affected by the uncertainties due a ¼ 105 eV=K; full circles: a ¼ 104 eV=K. The dashed lines are drawn to the thickness of the passive materials in the window as a guide for the eye. surface the silicon sensors. For instance, an underestima- tion of the thickness of these materials could decrease the 4 energy to create an e–h pair due to an over estimation of the photons ﬂux absorbed by the silicon. Vice versa, an 3.9 overestimation of the passive material thickness could increase the energy to create an e–h pair in silicon.

3.8

6.5 3.7 6 3.6 5.5

3.5 5 Pair Creation energy (eV) 3.4 4.5

4 3.3 3.5 3.2 2 3 1 10 10 10 Pair Creation energy (eV) 3 Photon Energy (eV) 2.5 Fig. 4. Electron–hole pair creation energy as a function of the incident photon energy. Full circles: T ¼ 140 K; open circles: T ¼ 300 K. The 2 dashed lines are drawn as a guide for the eye. 1.5 1 10 relation between W and E (i.e. one e–h pair per photon) is Photon Energy (eV) found at photon energies below 4 eV. This behavior is in Fig. 5. Electron–hole pair creation energy as a function of the incident agreement with experimental results reported in Ref. [12], photon energy at T ¼ 300 K. Full circles: standard simulation; open even though in Ref. [12] the maximum value is found at circles: case 1; open triangles: case 2. The dashed lines are drawn as a guide different photon energy, i.e. around 6 eV. for the eye. ARTICLE IN PRESS M.N. Mazziotta / Nuclear Instruments and Methods in Physics Research A 584 (2008) 436–439 439

References [6] F. Scholze, H. Rabus, G. Ulm, J. Appl. Phys. 84 (1998) 2926. [7] G.W. Fraser, et al., Nucl. Instr. and Meth. A 350 (1994) 368. [1] B.G. Lowe, R.A. Sareen, Nucl. Instr. and Meth. A 576 (2007) 367. [8] Y.P. Varshni, Physica 34 (1967) 149. [2] P. Lechner, L. Stru¨der, Nucl. Instr. and Meth. A 354 (1995) 464. [9] hhttp://www.ioffe.ru/SVA/NSM/Semicond/index.htmli. [3] M.N. Mazziotta, et al., Nucl. Instr. and Meth. A 533 (2004) 322. [10] V. Alex, S. Finkbeiner, J. Weber, J. Appl. Phys. 79 (1996) 6943. [4] R.C. Alig, S. Bloom, C.W. Struck, Phys. Rev. B 22 (1980) 5565. [11] K.P. O’Donnell, X. Chen, Appl. Phys. Lett. 58 (1991) 2924. [5] R.C. Alig, Phys. Rev. B 27 (1983) 968. [12] F. Scholze, et al., Nucl. Instr. and Meth. A 439 (2000) 208.