LSS theory

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The LSS theory (also Lindhard-Scharff-Schiøtt theory) is a description of the interaction of energetic ions with (amorphous) solids by Jens Lindhard , Morten Scharff and Hans E. Schiøtt . It is believed to be an improvement on the Bethe-Bloch formula for ions with energies in the kiloelectroenvolt range and is used, among other things, in calculating projected depth and other features of ion implantation . For higher energies and complex layer systems, the theory ( ZBL theory ) developed by James F. Ziegler , Jochen P. Biersack and Uffe Littmark should be used.

background

The beginnings of ion implantation for material investigation and modification go back to about 1955. After initial theories on the range of the penetrating particles and also on radiation damage to solids and even semiconductors, the theoretical basis for the range distribution of low-energy ions in solids was laid by Lindhard, Scharf and Schiøtt (LSS theory) in 1963, building on the work of Bohr.

description

However, this theory only describes the interaction of the ions with amorphous solids, i.e. That is, the lattice structure of a semiconductor crystal is not taken into account. The LSS theory cannot describe secondary effects such as diffusion of the implanted ions and generated defects either. The statements of the LSS theory are therefore only correct to a first approximation. The range of the ions plays a decisive role in ion implantation. In principle, five different braking mechanisms for ions are conceivable:

• inelastic collisions with bound electrons,
• inelastic collisions with atomic nuclei,
• elastic collisions with bound electrons,
• elastic collisions with atomic nuclei,
• Cherenkov radiation .

In practice, however, only the elastic collisions with atomic nuclei and the inelastic collisions with electrons are relevant.

Important parameters for describing the range of ions in the solid are the mean projected range, range spread, skewness and kurtosis ( more generally known in statistics as the first to fourth statistical moment of the probability density function):

Starting point for the derivation

The starting point for deriving the range description is the probability density function:

${\displaystyle \int _{-\infty }^{\infty }f(x)\mathrm {d} x=1}$ and ${\displaystyle f(x)\geq 0}$

as well as the general range distribution:

${\displaystyle C(x)=Nf(x)}$

N… implanted dose

Range and projected range of an ion in the solid

Mean projected range

The projected range of an ion describes the distance between the ion and the surface after it has come to rest. If one considers all implanted ions and forms the mean value of the projected ranges, one obtains the mean projected range . As a rule, this does not necessarily coincide with the location of the maximum concentration of the implanted ions. Mathematically, the mean projected range can be represented as follows: ${\displaystyle R_{\mathrm {p} }}$

${\displaystyle R_{p}=\int _{-\infty }^{\infty }xf(x)\mathrm {d} x}$

Range spread

The range spread describes the "breadth of the distribution" around the mean projected range. It can be described mathematically as follows: ${\displaystyle \Delta R_{\mathrm {p} }}$

${\displaystyle \Delta R_{\mathrm {p} }={\sqrt {\int _{-\infty }^{\infty }(x-R_{\mathrm {p} })^{2}f(x)\mathrm {d} x}}}$

Crookedness

The mean projected range and the range spread are suitable for describing symmetrical profiles. However, since implantation profiles are usually not symmetrical, two further sizes must be defined. One is the skewness, which indicates the asymmetry between the two areas "left and right" of the mean projected range. It can be stated mathematically as follows:

${\displaystyle \gamma ={\frac {\int _{-\infty }^{\infty }(x-R_{\mathrm {p} })^{3}f(x)\mathrm {d} x}{{\Delta \mathrm {R_{p}} }^{3}}}}$

Kurtosis

The second quantity is the kurtosis , which indicates the flatness of the maximum of the distribution:

${\displaystyle \beta ={\frac {\int _{-\infty }^{\infty }(x-R_{\mathrm {p} })^{4}f(x)\mathrm {d} x}{{\Delta R_{\mathrm {p} }}^{4}}}}$

literature

• Ingolf Ruge, Hermann Mader: Semiconductor technology . Springer, 1991, ISBN 3-540-53873-9 , pp. 100-104 .
• Frank Börner: Defect characterization in semiconductor layers with the help of positron annihilation . Halle-Wittenberg 2000 ( abstract and PDF - dissertation, mathematical-natural-scientific-technical faculty of the Martin-Luther-University Halle-Wittenberg).

Individual evidence

1. ^ ^{A b} J. Lindhard, M. Scharff, HE Schiøtt: Range concepts and heavy ion ranges (Notes on atomic collisions, II) . In: Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd . tape 33 , no. 14 , 1963, pp. 1-49 ( sdu.dk [PDF]).
2. ↑ ^{a b} N. Bohr: The decrease in velocity of alpha rays . In: Phil. Mag . tape 25 , 1913, pp. 10 .
3. ↑ N. Bohr: On the decrease of velocity of swiftly moving particles in passing through electrified matter . In: Phil. Mag . tape 30 , 1915, pp. 581-612 .
4. ^ RE Davis, WE Johnson, K. Lark-Horovitz, DS Siegel: Neutron bombarded Germanium semiconductors . In: Phys. Rev . tape 74 , 1948, pp. 1255 .