# The Valdinoci Temperature Dependent Impact Ionization Model for Silicon

Impact ionization plays an important role in the design of many semiconductor devices. Since impact ionization is both temperature and electric field dependent, it is important to use a model which has the correct functional behavior. For most devices impact ionization defines the safe region of operation within a moderate range of temperatures, so the temperature dependence of impact ionization is of lessor importance than the field dependence. However, for devices which must operate outside of the usual temperature limits, or which carry enough current to heat themselves out of those limits, both field dependence and temperature dependence of the model must be accurate.

Impact ionization has been measured and modeled extensively at room temperature. Recently, Valdinoci et al.[1], have extended the measured temperature range to 400°C and developed a compact model for both electron and hole impact ionization. Their analytic model is given by

_{n,p}= E/ ( a(T) + B(T) exp[d(T)/(E + c(T))])

where E is the elelctric field along the current flow lines. The parameters a ( ), b( ), c( ) and

d( ) are defined according to:

a(T) = a0 + a1*T

^{a2}b(T) = b0 * exp[b1*T]

c(T) = c0 + c1*T

^{c2}+ c3*T^{2}d(T) = d0 + d1*T + d

^{2}* T2

This model is now available in ** ATLAS**. It
is enabled by specifying VALDINOCI on the IMPACT statement. Valdinoci's
fitted parameters for silicon are the default and are shown in Table 1.
The model parameters for electrons and holes can be modified by specifying
the parameters VAL.AN0, VAL.AP0, etc., on the IMPACT statement.

In contrast to Valdinoci's model, the Selberherr impact ionization model is simpler:

= A(T) exp[-(B(T)/E)

^{beta}]

where the parameters A and B are temperature dependant according to

A(T) = A

_{1,2}(1+A.T[(TL/300)^{M.AT -1}])

and

B(T) = B

_{1,2}(1+B.T[(TL/300)^{M.BT -1}])

A1 and B1 are used for E < EGRAN; A2 and B2 are used for E > EGRAN. (Note that the Valdinoci model reduces to the Selberherr model when its parameters are set to zero except for b0 and d0. But simpler is not necessarily more accurate.)

The electron or hole impact ionization coefficients at a location in a device can be recorded in a log file by specifying ALPHAN or ALPHAP in the PROBE statement. Figure 1 and Figure 2 each show a family of plots of electron ionization coefficient versus 1/E for the Selberherr model and the Valdinoci model. The Selberherr model gives straight lines (since beta=1 for silicon in this case) while the Valdinoci model shows a more complicated behavior. Figure 3 and Figure 4 compare the Selberherr and Valdinoci models directly at 300K and 700K. At 300K, both models give similar results, but at 700K the Selberherr model will underestimate low field impact ionization effects. In summary, the Valdinoci model is recommended for silicon devices which experience an extended temperature range of operation.

Figure 1. Variation of the electron
ionization rate from

Selberherr model, with inverse electric field

for varying ambient temperatures.

Figure 2. Variation of the electron
rate from Valdinoci’s

model, with the inverse electric field

for varying ambient temperatures.

Figure 3. Comparison between the Valdinoci

and Selberher models of electron ionization

rate at an ambient temperature of 300k.

Figure 4. Comparison between the Valdinoci
and

Selberher model of electron ionization

rate at an ambient temperature of 700k.

Parameters | Electrons |
Holes |
---|---|---|

a_{0} |
4-34 |
2.37 |

a_{1} |
-2.4x10 ^{-12} |
0.01 |

a_{2} |
4.12 |
1 |

b_{0} |
0.235 |
0.177 |

b_{1} |
0 |
-0.002 |

c_{0} |
1.68x10 ^{4} |
0 |

c_{1} |
4.38 |
0.09 |

c_{2} |
1 |
2.5 |

c_{3} |
0-13 |
0 |

d_{0} |
1.23x10 ^{6} |
1.4x10 ^{6} |

d_{1} |
1.2x10 ^{3} |
2.97x10 ^{3} |

d_{0} |
0.567 |
1.48 |

**References**

- M. Valdinoci, D. Ventura, M. C. Vecchi, M. Rudan, G. Baccarani, F. Illien, A. Stricker, L. Zullino, "Impact-ionization in silicon at large operating temperature", SISPAD '99, Sept. 6-8, 1999, Kyoto, Japan.