The theory of Fan describes the thermally provoked shift of the energy bandgap of semiconductors. Part of the theory is the Fan factor, which depends on the microscopic material parameters, such as the dielectric constants and effective masses. Herein, employing data of GaP, we show that the originally published Fan factor was missing one multiplicative numerical factor, which turned out to be the high frequency dielectric constant.

In the early 1950s, Fan^{1} introduced a theory describing the thermally evoked variation of the energy bandgap (*E*_{g}) in semiconductors. Fan’s formula,

accurately describes the thermodynamics of the affair,^{2} where *E*_{0} is the bandgap energy at zero Kelvin, *A* is the Fan factor, ⟨*E*_{LO}⟩ is the average longitudinal optical (LO) phonon energy responsible for the *E*_{g} shift, *k* is the Boltzmann constant, and *T* is the ambient temperature. In a follow-up paper,^{3} Fan introduced the expression for *A*, i.e., Eq. (6.3) in Ref. 3, while herein, using SI units, *e*^{2} is replaced by *e*^{2}/(4*πε*_{0}), resulting in the following expression:

where *e* (=1.602 × 10^{−19} As) is the elementary charge, *ℏ* (=1.055 × 10^{−34} Js) is the reduced Planck constant, *m*_{0} (=9.11 × 10^{−31} kg) is the free electron mass, *ε*_{0} (=8.854 × 10^{−12} As V^{−1} m^{−1}) is the vacuum permittivity, *ε*_{∞} is the high frequency, *ε*_{s} is the static dielectric constant, and $mc*$ and $mv*$ are the effective mass of the electrons and holes, respectively, in multiples of *m*_{0}.

The symbols, which have been retrieved from Ref. 4, and the solid line in Fig. 1 show the measured shift of *E*_{g} vs temperature for the III–V compound semiconductor GaP and the fit with Eq. (1), respectively. The fit parameters are listed in Table I. Notably, ⟨*E*_{LO}⟩ = 0.050 eV excellently matches the GaP LO phonon energy of 0.049 eV in the literature.^{5,6} However, in the course of our recent work,^{2} despite the excellent fit (goodness of fit > 0.99) revealed in Fig. 1 and the harmony of the fitted LO phonon energy with the literature, for various semiconductors, we recognized persistent inconsistencies between the *A* values attained with the fit using Eq. (1) and the theoretically expected ones calculated with Eq. (2). In order to clearly demonstrate the misfit, we use the well-established LO phonon energy. Rearranging Eq. (2), ⟨*E*_{LO}⟩ is given by

By inserting the fit result *A* = 0.340 eV and the dielectric constants^{5} and effective masses of GaP,^{6} which are shown in Table II, into Eq. (3), we expected to find ⟨*E*_{LO}⟩ ∼ 0.050 eV. However, the calculation results in ⟨*E*_{LO}⟩ = 7.1 eV, a number two orders of magnitude larger.

E_{0} (eV)
. | ⟨E_{LO}⟩ (eV)
. | A (eV)
. |
---|---|---|

2.328 ± 0.007 | 0.050 ± 0.006 | 0.340 ± 0.049 |

E_{0} (eV)
. | ⟨E_{LO}⟩ (eV)
. | A (eV)
. |
---|---|---|

2.328 ± 0.007 | 0.050 ± 0.006 | 0.340 ± 0.049 |

ε_{s}
. | ε_{∞}
. | $mc*$ . | $mv*$ . |
---|---|---|---|

11.0^{5} | 8.8^{5} | 0.35^{6} | 0.86^{6} |

ε_{s}
. | ε_{∞}
. | $mc*$ . | $mv*$ . |
---|---|---|---|

11.0^{5} | 8.8^{5} | 0.35^{6} | 0.86^{6} |

Because of the clear discrepancy, we revisited Ref. 1. For polar semiconductors, such as the III–V compound GaP, Eq. (35) therein defines *A* as follows:

where *e*^{*} is the effective charge of the ions, *a* is the interionic distance, and *M* is the ion mass.

Rearranging Eqs. (32) and (31) in Ref. 1 gives

and

which transfers Eq. (4) into

whereas with ⟨*E*_{LO}⟩ = *ℏω*_{LO}, it follows for SI units that

and

which corrects the result of Eq. (3) to *E*_{LO} = 0.092 eV. The number is in fair agreement with the expected outcome of *E*_{LO} ∼ 0.050 eV. Equivalently, we calculated the *A* values. The results of Eqs. (2), (3), (9), and (10) are displayed in Table III.

Using the expression for the fine structure constant *α* = *e*^{2}/(4*πℏcε*_{0}), where *c* (= 2.998 × 10^{8} m/s) is the speed of light, reduces the parameters in Eqs. (9) and (10), resulting in

and

respectively. Finally, with the Rydberg energy |*E*_{Ryd}| = (*m*_{0}*c*^{2}*α*^{2})/2 = 13.6 eV, we express *A* with the straightforward numerical value expression,

where ⟨*E*_{LO}⟩ is the LO phonon energy in eV.

The finding herein causes consequences for our own former work. In Refs. 7 and 8, we used Eq. (2) to calculate the intrinsic Stokes shift (Δ_{Stokes}) and Huang–Rhys factor (*S*) for GaAs and CdS. For GaAs, the corrected result is Δ_{Stokes} = 4.2 meV × 11.6 = 48.7 meV, and for CdS, it is *S* = 1.75 × 5.23 = 9.2, where 4.2 meV and 1.75 are the calculated results in Refs. 7 and 8, which ought to be multiplied with the corresponding values of *ε*_{∞}, i.e., 11.6 and 5.23. Further discussions in the light of the current work of our previous results require the inclusion of the uncertainties of the experimental data and material parameters and shall be presented in a future work.

Summarizing, represented by Eq. (9), we derived the correct formula for the Fan factor *A*, producing values satisfactorily matching the fit results gained from Eq. (1). Additionally, this work introduces a straightforward numerical value expression for the calculation of the Fan factor of polar semiconductors.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.