# SHEET RESISTANCE CALCULATOR

This calculator determines the sheet resistance of an arbitrarily doped semiconductor at equilibrium (i.e., in the dark and with no applied voltage).

The calculator simulates a four-point probe measurement of a surface diffusion, such as the emitter or the back-surface field of a photovoltaic solar cell. The user can either generate a dopant profile, or upload a profile from a SIMS, ECV, or spreading-resistance measurement. The calculator then determines the sheet resistance and the junction depth of the surface diffusion at any temperature.

The sheet resistance *ρ*_{sq} at equilibrium is determined from the net ionised doping concentration *N*(*z*) and the mobility of the majority carriers *μ*_{maj} by the equation

where *z*_{j} is the junction depth and *q* is the charge of an electron. The sheet resistance has the dimensions Ω/sq.

The net ionised doping concentration is defined as *N*(*z*) = |*N*_{A}(*z*) – *N*_{D}(*z*)|, where *N*_{A} and *N*_{D} are the ionised concentration of acceptor and donor atoms. In the case of a silicon semiconductor, boron atoms are acceptors and phosphorus atoms are donors.

A doping profile can be uploaded from a CSV file or generated as an exponential, Gaussian or inverse error function (ERFC). The equations for these functions are shown below. The doping profile represents a surface diffusion, such as an emitter or a back-surface field. These are usually created by diffusing dopant atoms into the semiconductor at high temperatures, but can also be created by implanting dopant atoms at high energies. A background concentration can be included in the analysis, which is necessarily a uniform doping profile.

The junction depth *z*_{j} is defined as follows: If there is no background, *z*_{j} equals the depth at which the minimum permissible dopant concentration *N*_{min} occurs, as defined on the 'Option' page; if the background and dopant profile are of opposite types, *z*_{j} is the depth at which |*N*_{A}(*z*) – *N*_{D}(*z*)| equals zero; and if the background and dopant profiles are of the same type, *z*_{j} equals the background thickness. The latter two definitions were chosen so that the simulator returns the sheet resistance that would be measured by an ideal four-point probe measurement.

The user has the option of several mobility models, which are described on the About page of the mobility calculator

**Warning:** The input profile is the *ionised* dopant profile. It should not include inactive dopant atoms, like interstitial atoms, or non-ionised substitutional atoms, which can occur when the dopant concentration is high or the temperature low.

# EQUATIONS FOR THE DOPING PROFILE

Like PC1D, the user can select from four equations where all are defined by a peak concentration *N*_{p}, the depth of the peak concentration *z*_{p}, and a depth factor *z*_{f}.

PC1D defines *z*_{f} such that the Gaussian and ERFC equations have the simplest form.

Uniform:^{ } |
*N*(*z*) = *N*_{p} for *z* ≤ *z*_{p} and *N*(*z*) = 0 for *z* > *z*_{p} |

Exponential:^{ } |
*N*(*z*) = *N*_{p} ⋅ exp[–|*z* – *z*_{p}| / *z*_{f}] |

Gaussian: |
*N*(*z*) = *N*_{p} ⋅ exp[–(*z* – *z*_{p})^{2} / *z*_{f}^{2} ] |

ERFC:^{ } |
*N*(*z*) = *N*_{p} ⋅ {1 – erf[ (*z* – *z*_{p}) / *z*_{f} ] } |

The traditional approach to employing Gaussian and ERFC equations is to set the standard deviation as the third input parameter. In this calculator, the user can select to define *z*_{f} as the standard deviation, in which case the profiles are defined as follows.

Uniform:^{ } |
*N*(*z*) = *N*_{p} for *z* ≤ *z*_{p} and *N*(*z*) = 0 for *z* > *z*_{p} |

Exponential:^{ } |
*N*(*z*) = *N*_{p} ⋅ exp[–|*z* – *z*_{p}| / *z*_{f}] |

Gaussian: |
*N*(*z*) = *N*_{p} ⋅ exp{–[ (*z* – *z*_{p}) / (*z*_{f} ⋅ √2) ]^{2}} / [*z*_{f} ⋅ √(2π)] |

ERFC:^{ } |
*N*(*z*) = *N*_{p} ⋅ {1 – erf[ (*z* – *z*_{p}) / (*z*_{f} ⋅ √2) ] } |

The selection of either the PC1D definition or the traditional definition of *z*_{f} is now available on the 'Calculator' page. Given the widespread use of PC1D in the field of photovoltaics, the default definition of *z*_{f} follows the PC1D definition.

# ACTIVE DOPING LIMIT

A maximum limit to the electrically active dopant concentration can be assigned. This maximum limit can be used to represent the solid solubility limit of the dopant species at a particular process temperature.

A future update of this calculator will include the solid solubility models from the literature.

# MOBILITY MODELS

For c-Si, the user may choose from the mobility models of Klaassen [1,2], Arora [3] or Dorkel and Leturcq [4].

For more information on the use of these models we suggest you visit either the Mobility Calculator or the Resistivity Calculator.

# ACKNOWLEDGEMENTS

We thank Fa-Jun Ma (SERIS) for finding a bug that caused an underestimation of the sheet resistance for B diffusions in versions preceding 1.4 (12-Apr-2013); and Kean Fong (ANU) for finding a bug that caused an overestimation of the sheet resistance for 'traditional' Gaussian profiles in versions preceding 1.6 (4-Dec-2013). We also thank Dongchul Suh (ANU) and Yimao Wan (ANU), for the Korean and Chinese translations.

# REFERENCES

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[1] | D.B.M. Klaassen, "A unified mobility model for device simulation—I. Model equations and concentration dependence," *Solid-State Electronics*, **35** (7), pp. 953–959, 1992. |

[2] | D.B.M. Klaassen, "A unified mobility model for device simulation—II. Temperature dependence of carrier mobility and lifetime," *Solid-State Electronics*, **35** (7), pp. 961–967, 1992. |

[3] | N.D. Arora, J.R. Hauser and D.J. Roulston, "Electron and hole mobilities in silicon as a function of concentration and temperature," *IEEE Transactions on Electron Devicies*, **ED-29** (2), pp. 292–295, 1982. |

[4] | J.M. Dorkel and PH. Leturcq, "Carrier mobilities in silicon semi-empirically related to temperature, doping and injection level," *Solid-State Electronics*, **24** (9), pp. 821–825, 1981. |

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