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, Fan1 introduced a theory describing the thermally evoked variation of the energy bandgap (Eg) in semiconductors. Fan’s formula,

$Eg=E0−Aexp(ELO/kT)−1,$
(1)

accurately describes the thermodynamics of the affair,2 where E0 is the bandgap energy at zero Kelvin, A is the Fan factor, ⟨ELO⟩ is the average longitudinal optical (LO) phonon energy responsible for the Eg 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, e2 is replaced by e2/(4πε0), resulting in the following expression:

$A=12ℏm0ELO12e24πε01ε∞−1εsmc*m012+mv*m012,$
(2)

where e (=1.602 × 10−19 As) is the elementary charge, (=1.055 × 10−34 Js) is the reduced Planck constant, m0 (=9.11 × 10−31 kg) is the free electron mass, ε0 (=8.854 × 10−12 As V−1 m−1) is the vacuum permittivity, ε is the high frequency, εs is the static dielectric constant, and $mc*$ and $mv*$ are the effective mass of the electrons and holes, respectively, in multiples of m0.

The symbols, which have been retrieved from Ref. 4, and the solid line in Fig. 1 show the measured shift of Eg 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, ⟨ELO⟩ = 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), ⟨ELO⟩ is given by

$ELO=1m04π2ℏε0Ae21ε∞−1εsmc*m012+mv*m012−12.$
(3)

By inserting the fit result A = 0.340 eV and the dielectric constants5 and effective masses of GaP,6 which are shown in Table II, into Eq. (3), we expected to find ⟨ELO⟩ ∼ 0.050 eV. However, the calculation results in ⟨ELO⟩ = 7.1 eV, a number two orders of magnitude larger.

FIG. 1.

The Eg variation of GaP vs temperature. The symbols, which have been retrieved from Ref. 4, represent the experimental data, and the solid line represents the fit using Eq. (1).

FIG. 1.

The Eg variation of GaP vs temperature. The symbols, which have been retrieved from Ref. 4, represent the experimental data, and the solid line represents the fit using Eq. (1).

Close modal
TABLE I.

Fit parameters in Eq. (1).

E0 (eV)ELO⟩ (eV)A (eV)
2.328 ± 0.007 0.050 ± 0.006 0.340 ± 0.049
E0 (eV)ELO⟩ (eV)A (eV)
2.328 ± 0.007 0.050 ± 0.006 0.340 ± 0.049
TABLE II.

GaP parameters used in Eqs. (2), (3), (9), and (10).

εsε$mc*$$mv*$
11.05 8.85 0.356 0.866
εsε$mc*$$mv*$
11.05 8.85 0.356 0.866

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:

$A=πe*2a3Me2ℏ12ωLO322mc*ℏ212+2mv*ℏ212,$
(4)

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

$πe*2a3M=εs−ε∞2ε∞ωTO2$
(5)

and

$ωTO2=ε∞εsωLO2,$
(6)

respectively, and substituting Eq. (6) in Eq. (5) results in

$πe*2a3M=εs−ε∞2εsωLO2,$
(7)

which transfers Eq. (4) into

$A=εs−ε∞2εs(ℏωLO)12e2ℏmc*12+mv*12,$
(8)

whereas with ⟨ELO⟩ = ℏωLO, it follows for SI units that

$A=12ℏm0ELO12e24πε0εs−ε∞εsmc*m012+mv*m012$
(9)

and

$ELO=1m04π2ℏε0Ae2εs−ε∞εsmc*m012+mv*m012−12,$
(10)

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

TABLE III.

Calculated values of A and ⟨ELO⟩ using Eqs. (2), (3), (9), and (10).

A (eV)ELO⟩ (eV)A (eV)ELO⟩ (eV)
[Eq. (2)][Eq. (3)][Eq. (9)][Eq. (10)]
0.028 7.1 0.250 0.092
A (eV)ELO⟩ (eV)A (eV)ELO⟩ (eV)
[Eq. (2)][Eq. (3)][Eq. (9)][Eq. (10)]
0.028 7.1 0.250 0.092

Using the expression for the fine structure constant α = e2/(4πℏcε0), where c (= 2.998 × 108 m/s) is the speed of light, reduces the parameters in Eqs. (9) and (10), resulting in

$A=α2m0c2ELO12εs−ε∞εsmc*m012+mv*m012$
(11)

and

$ELO=A2m0c22αεs−ε∞εsmc*m012+mv*m012−12,$
(12)

respectively. Finally, with the Rydberg energy |ERyd| = (m0c2α2)/2 = 13.6 eV, we express A with the straightforward numerical value expression,

$A=13.6×ELO1/2εs−ε∞εsmc*m012+mv*m012,$
(13)

where ⟨ELO⟩ 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 (ΔStokes) and Huang–Rhys factor (S) for GaAs and CdS. For GaAs, the corrected result is ΔStokes = 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.

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

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