# MOBILITY CALCULATOR

The user selects (i) the semiconductor material (c-Si), (ii) the dopant species (P or B), (iii) the **substitutional** dopant concentration *N*_{dop},
(iv) the excess carrier concentrations Δ*n*, Δ*p*, and (v) the temperature *T*.

The program calculates the following outputs: the mobility of electrons *μ*_{e} and holes *μ*_{p};
the ambipolar mobility *μ*_{a}; the equivalent carrier diffusivities, *D*_{e}, *D*_{h} and *D*_{a};
the equivalent diffusion lengths, *L*_{e}, *L*_{h} and *L*_{a}, for user-defined effective carrier lifetimes,
*τ*_{eff e} and *τ*_{eff h}; and the semiconductor's resitivity in equilibrium *ρ*_{0} and steady-state *ρ*.
The outputs can be plotted as a function of *N*_{dop}, Δ*n* = Δ*p*, or *T*.

# Equations

The equilibrium carrier concentrations are given by

*n*_{0} = *N*_{ionised} and *p*_{0} = *n*_{i}^{2}/*N*_{ionised} for an n-type semiconductor, and

*p*_{0} = *N*_{ionised} and *n*_{0} = *n*_{i}^{2}/*N*_{ionised} for a p-type semiconductor,

where *n*_{i} is the intrinsic carrier concentration. It is determined from user-selected models for the semiconductor
bandgap and density of states. Note that the calculator also converts the substitutional dopant concentration *N*_{dop} to the ionised dopant concentration *N*_{ionised} using the selected ionisation model.

The steady-state carrier concentrations are given by

*n* = *n*_{0} + Δ*n*, and

*p* = *p*_{0} + Δ*p*.

The carrier mobilities, *μ*_{e} and *μ*_{h}, are determined by following a user-selected mobility model,
where the mobilities depend on *T*, *n*_{0}, *p*_{0}, *n* and *p*. Additional detail on these models is given below.

The remaining outputs are calculated as

*μ*_{a} = (*n* + *p*)⋅*μ*_{e}⋅*μ*_{h} / (*n*⋅*μ*_{e} + *p*⋅*μ*_{h}),

*D*_{m} = *μ*_{m} ⋅ *k*⋅*T*/*q*,

*L*_{m} = √(*D*_{m}⋅*τ*_{m}),

*ρ*_{0} = 1 / [*q*⋅(*μ*_{e}_{0}⋅*n*_{0} + *μ*_{h}_{0}⋅*p*_{0})], and

*ρ* = 1 / [*q*⋅(*μ*_{e}⋅*n* + *μ*_{h}⋅*p*)],

where *q* is the charge of an electron, *k* is Boltzmann's constant, and the subscript *m* represents
*e*, *h* or *a*.

# Mobility models

For c-Si, the user may choose from the mobility models of Schindler *et al.* [1], Klaassen [2, 3] and Dorkel and Leturcq [5]. There exist several other
mobility models for c-Si that are limited to equilibrium because they do not account for non-zero excess carrier concentrations. A calculator that
determines the carrier mobilities at equilibirium only—and thereby permits
the selection of alternative mobility models—can be found here.

It is recommended that the user select either Klaassen's or Schindler's mobility model. The calculations for Klaasen's model follow the equations and procedure presented in [2, 3]
with two exceptions: (i) *r*_{5} is set to –0.8552 rather than –0.01552 (see Table 2 of [2]), and (ii) Eq. A3 of [2] is adjusted such that *P*_{CWe}
is determined with *N*_{e,sc} rather than (*Z*_{D}^{3}⋅*N*_{I}) and *P*_{CWh} is determined with *N*_{h,sc} rather than
(*Z*_{A}^{3}⋅*N*_{I}); these changes give a better fit to the solid calculated lines in Figures 6 and 7 of [2], which better fits the experimental data.
These modifications are also contained in Sentaurus's version of Klaassen's model [6]. Klaassen's mobility model fits reasonably with experimental data
over an estimated temperature range of 100–450 K, where its accuracy is greatest at 300 K (see [2,3]).

If the model of Dorkel and Leturcq is selected, the calculations follow Equation 7 in [5], but where the value of 2e7 has been corrected to 2e17 to make it consistent with
Equation 3 in [5] and to give positive mobilities. Dorkel and Leturcq state that the calculated electron mobility deviates by an average of 5% from experimental values,
except at low T (~200 K) where it is inaccurate; and that the calculated hole mobility at 300 K is within 5% of experimental values except at doping concentrations in excess
of 2e18 cm^{–3}. At the time of Dorkel and Leturcq's work, there was little experimental data with which to compare hole mobilities at temperatures other
than 300 K.

# Other physical models

The program provides the user the choice of several physical models from which the intrinsic bandgap and density of states—and hence the
intrinsic carrier concentration *n*_{i}—can be calculated. More information on these models is provided in other calculators.

# REFERENCES

| |

[1] | F. Schindler, M. Forster, J. Broisch, J. Schön, J. Giesecke, S. Rein, W. Warta and M.C. Schubert, "Towards a unified low-field model for carrier mobilities in crystalline silicon," *Solar Energy Materials and Solar Cells*, **131**, pp. 92–99, 2014. |

[2] | 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. |

[3] | 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. |

[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. |

[5] | SENTAURUS, Synopsys Inc. Mountain View, CA, www.synopsys.com/products/tcad/tcad.html. |

# ACKNOWLEDGEMENTS

Thanks to Pietro Altermatt from the University of Hannover, Achim Kimmerle, Andreas Wolf and Florian Sch from Fraunhofer ISE, Helmut Maekel from Centrotherm, and Fiacre Rougieux from the Australian National University for assistance and discussion on interpreting mobility models.

# FEEDBACK

Please email corrections, comments or suggestions to support@pvlighthouse.com.au.
We would appreciate receiving references that relate to the resistivity and carrier mobility in semiconductors other than c-Si.