Physical models
Carrier mobility	Constant mobilities Schindler2014 Klaassen1992 DorkelLeturcq1981
Dopant ionisation	100% ionisation Altermatt2006
Intrinsic band gap	Passler2002 Bludau1974–Green1990 Sentaurus2008		Band gap multiplier
Density of states	Green1990 (Eqs 4–6,8,9,13) Green1990 (Eqs 4,5,14,15) Sentaurus2008 DOS Form. 1 Sentaurus2008 DOS Form. 2 Couderc2014 (Eqs 4,5,18,19)
Carrier statistics	Boltzmann Fermi–Dirac
Band gap narrowing	None Yan–Cuevas2013 Schenk1998 delAlamo1985

Miscellaneous
Significant figures in outputs
Force equal excess carrier concentrations, Δn = Δp
Force equal carrier lifetimes, τeff e = τeff h
Plot figure
Include ambipolar

ρ₀, ρ, μ_e, μ_h, μ_a, D_e, D_h and D_a at steady-state for boron-doped c-silicon applying the Klaassen1992 mobility model.

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 Δn, Δp, 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 ρ₀ and steady-state ρ. The outputs can be plotted as a function of N_dop, Δn = Δp, or T.

The equilibrium carrier concentrations are given by

n₀ = N_ionised and p₀ = n_i²/N_ionised for an n-type semiconductor, and

p₀ = N_ionised and n₀ = n_i²/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 = n₀ + Δn, and

p = p₀ + Δp.

The carrier mobilities, μ_e and μ_h, are determined by following a user-selected mobility model, where the mobilities depend on T, n₀, p₀, 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),

ρ₀ = 1 / [q⋅(μ_e0⋅n₀ + μ_h0⋅p₀)], 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.

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) r₅ 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_D³⋅N_I) and P_CWh is determined with N_h,sc rather than (Z_A³⋅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.

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.

[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.

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.

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.