Journal of Physical Science and Application 2 (8) (2012) 269-273

Modeling of Track Formation in Semiconductors Irradiated with Swift Heavy Ions

Soraya Kadid and Ali Meftah

Laboratoire de Recherche en Physico-Chimie des Surfaces et Interfaces, Département des Sciences de la Matière, Faculté des

Sciences, Université 20 Août 1955 Skikda, Route d’El Hadaiek, BP 26, 21000 Skikda, Algérie Received: May 19, 2012 / Accepted: June 06, 2012 / Published: August 15, 2012. Abstract: The interaction of the heavy charged particles, of energy higher than a few MeV/amu with semiconductor single crystals can lead to the structural modification of their physical properties and participate at the creation of the defects which are called latent tracks. Several models were tested for explaining the track formation in semiconductors irradiated with swift heavy ions, one of them is the thermal spike model. This work shows that the experimental data obtained in semiconductors, in our case in InP irradiated with swift heavy ions can be described on the basis of the thermal spike model. The experimental results obtained on InP have allowed the parameters of this model to be understood. The only free parameter is the electron-phonon coupling constant g which is unknown in InP. This model allows the evolution of track radii to be found as a function of electronic stopping power (dE/dx)e for different beam energies. For InP a good agreement is observed between calculated track radii and experimental ones on one hand, and on the other hand between calculated and experimental threshold value of electronic stopping power. This allows determining the electron-phonon coupling value for InP to be equal 0.9 × 1011 W·cm-3·K-1 and the (dE/dx)e threshold for latent track formation in InP equal 27 ± 3 keV/nm for ion energies ranging from 0.4-10 MeV/amu. Key words: Semiconductors, swift heavy ions, ion tracks, thermal spike model.

1. Introduction

The study of interaction of swift heavy ions with semiconductor single crystals is very important both for the fundamental investigations of radiation effects in condensed matter and for the creation of ion tracks in semiconductor materials, which can be used in modern nanotechnologies of electronics. Several theoretical models have been proposed in order to explain the appearance of latent tracks induced in matter by the slowing down process of incident ions in the electronic stopping power regime. Two of the most developed models are based on either the thermal spike approach [1, 2] or the coulomb explosive spike mechanism [3, 4]. The thermal spike model assumes that the electronic excitation energy is first transferred to the atoms

Corresponding author: Soraya Kadid, Ph.D., research field:

interaction of swift heavy ions with materials. E-mail: Kadidsora2000@yahoo.fr.

followed by a strong local rise of the lattice temperature up to (and higher than) the melting point. In this case, a latent track results from rapid solidification of a cylinder of molten matter [1, 2]. The present paper aims at proposing a detailed approach of the use of the thermal spike model in order to describe quantitatively the amorphous track radii induced in crystalline semiconductors by swift heavy ions. The knowledge of experimental track radii of a given semiconductor, in our study InP allows determining the electron-phonon coupling constant g which is unknown in this material.

2. The Thermal Spike Model

2.1 Mathematical Description

The thermal spike model is described mathematically by two coupled equations [5] governing the energy diffusion into the electron

DAVID PUBLISHING

D

Modeling of Track Formation in Semiconductors Irradiated with Swift Heavy Ions

270

subsystem and into the lattice subsystem. A time, dependent transient thermal process

coupling these two systems is expressed using a cylindrical geometry whose axis is the ion path:

),()())((1 trAgr

rKrrt

C aee

eee

e +Τ−Τ−∂Τ∂

Τ∂∂

=∂Τ∂ (1)

)())((1ae

aaa

aa g

rrK

rrtC Τ−Τ+

∂Τ∂

Τ∂∂

=∂Τ∂ (2)

Ci, Ki and Ti are the specific heat, the thermal conductivity and the temperature, respectively, i is referring either to the electrons (e) or to the lattice atoms (a), g is the electron-phonon interaction [5] and A(r, t) is the energy density deposited by an incident ion in the target electrons.

3. Results and Discussion

Eqs. (1) and (2) are solved numerically. In our calculations we will lay emphasis on the discussion of the key parameter: the electron-phonon constant g in crystalline semiconductors. The main calculation results for different experimental conditions (ion species and irradiation temperature) are depicted in Figs. 1 and 2 where the electron and lattice temperatures are plotted versus time for g = 0.9 × 1011 W·cm-3·K-1 and with a beam energy of 5.7 MeV/amu.

3.1 Track Radii and Comparison with Experiments

In this work, this model was applied to calculate

Fig. 1 Evolution of the electronic temperature versus time at several radii from the ion path for g = 0.9 × 1011 W·cm-3·K-1. The initial conditions are: beam energy 5.7 MeV/amu, initial temperature 300 K.

Fig. 2 Evolution of the lattice temperature versus time at several radii from the ion path for g = 0.9 × 1011 W·cm-3·K-1. The initial conditions are: beam energy 5.7 MeV/amu, initial temperature 300 K.

0 1x10-12 2x10-12 3x10-120

500

1000

1500

Tm

Radius (nm)

Latti

ce te

mp閞

atur

e (k

)

Time (s)

1 2 3 4 5 6 7 8 9 10 11

0 2x10-14 4x10-14 6x10-140

50000

100000

150000

Radius (nm)

Elec

tron

ic te

mp閞

atur

e (K

)

Time (s)

1 2 3 4 5 6 7 8 9 10 11

Elec

troni

c te

mpe

ratu

re (K

) La

ttice

tem

pera

ture

(k)

Modeling of Track Formation in Semiconductors Irradiated with Swift Heavy Ions

271

track radii versus electronic stopping power and the electronic stopping power threshold. Figs. 3 and 4 show the latent track radius evolution versus (dE/dx)e for two different values of the electron-phonon coupling g = 0.9 × 1011 W·cm-3·K-1 and 1.8 × 1011

W·cm-3·K-1 and two different beam energies (5.7 and 10 MeV/amu). In these figures, the comparison between the calculated radii and the experimental ones is given for InP. The Rexp values are taken from Refs. [6-8]. In a first approach the best fit is realized

for g = 0.9 × 1011 W·cm-3·K-1 assuming that the formation of a latent track results of a rapid quench of the melt phase. The threshold value for melting equal 27 ± 3 keV/nm not far from the experimental result (20-25 keV/nm) [9, 10].

To confirm our results, Figs. 5 and 6 show the good agreement, between our calculations and the experimental data for g = 0.9 × 1011 W·cm-3·K-1. Therefore, this value is a reasonable estimate value of the electron-phonon coupling in InP.

Fig. 3 Latent track radii Rt (nm) in InP at 300 K as a function of the electronic stopping power (dE/dx) keV/nm. The experimental points correspond to various ion energies in the range 0.4-10 MeV/amu. The calculated curves have been plotted for two values of the electron phonon coupling g at the beam energy E = 5.7 MeV/amu.

Fig. 4 Latent track radii Rt (nm) in InP at 300 K as a function of the electronic stopping power (dE/dx) keV/nm. The experimental points correspond to various ion energies in the range 0.4-10 MeV/amu. The calculated curves have been plotted for two values of the electron phonon coupling g at the beam energy E = 10 MeV/amu.

0 5 10 15 20 25 30 35 40

2

4

6

8

10

12

Trac

k ra

dius

Rt(n

m)

Electonic Stopping Power (dE/dx)e(keV/nm)

Exp Data--- Calc g=1.8X1012Wcm-3K-1

Calc g=0.9x1011 Wcm-3 K-1

0 5 10 15 20 25 30 35 40

2

4

6

8

10

12

Trac

k ra

dius

Rt (

nm)

Electronic stopping power (dE/dx)e (keV/nm)

Exp Data--- Calc g=1.8x1011 Wcm-3K-1

Calc g=0.9x1011 Wcm-3 K-1)

Modeling of Track Formation in Semiconductors Irradiated with Swift Heavy Ions

272

0 5 10 15 20 25 30 35 40

2

4

6

8

10

12

Trac

k ra

dius

Rt(n

m)

Electronic Stopping Power (dE/dx)e (KeV/nm)

Exp Data Calc 5.7 Mev/amu----Calc10 MeV/amu

Fig. 5 Latent track radii Rt (nm) in InP at 300 K as a function of the electronic stopping power (dE/dx)e. The experimental points correspond to various ion energies in the range 0.4-10 MeV/amu. The calculated curves have been plotted for g = 1.8 × 1011 W·cm-3·K-1.

0 5 10 15 20 25 30 35 40

2

4

6

8

Tra

ck ra

dius

Rt(n

m)

Electronic Stopping Power (dE/dx)e (keV/nm)

Exp Data Calc5.7 MeV/amu--- Calc10MeV/amu

Fig. 6 Latent track radii Rt (nm) in InP at 300 K as a function of the electronic stopping power (dE/dx)e. The experimental points correspond to various ion energies in the range 0.4-10 MeV/amu. The calculated curves have been plotted for g = 0.9 × 1011 W·cm-3·K-1.

5. Conclusions

It is shown that the experimental data obtained for InP can be described by the thermal spike model. The most important input parameter: the electron-phonon coupling constant g, is determined from available experimental data on ion track formation in InP. The calculated track radii vary with ion energy E and thus with (dE/dx)e; further, they are reasonable and correlate well with experimental results if g is equal to 0.9 × 1011 W·cm-3·K-1. The threshold value deduced for melting is equal to 27 ± 3 keV/nm. Within the

measurement errors near, this value coincides with the experimental results published. Our results indicate that the thermal spike model can adequately describe the track formation in semiconductor crystals irradiated by swift heavy ions

Acknowledgments

The authors would like to thank Dr. M. Marcel (CIMAP, Caen, France) for useful discussions.

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