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For an electronic system, given a mean field method and a distribution of orbital occupation numbers that are close to the natural occupations of the correlated system, we provide formal evidence and computational support to the hypothesis that the entropy (or more precisely −σS, where σ is a parameter and S is the entropy) of such a distribution is a good approximation to the correlation energy. Underpinning the formal evidence are mild assumptions: the correlation energy is strictly a functional of the occupation numbers, and the occupation numbers derive from an invertible distribution. Computational support centers around employing different mean field methods and occupation number distributions (Fermi–Dirac, Gaussian, and linear), for which our claims are verified for a series of pilot calculations involving bond breaking and chemical reactions. This work establishes a formal footing for those methods employing entropy as a measure of electronic correlation energy (e.g., iDMFT [Wang and Baerends, Phys. Rev. Lett. 128, 013001 (2022)] and TAODFT [J.D. Chai, J. Chem. Phys. 136, 154104 (2012)]) and sets the stage for the widespread use of entropy functionals for approximating the (static) electronic correlation.
more » « less NSFPAR ID:
 10494874
 Publisher / Repository:
 American Institute of Physics
 Date Published:
 Journal Name:
 The Journal of Chemical Physics
 Volume:
 159
 Issue:
 19
 ISSN:
 00219606
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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A 15μmdiammesa device was defined by standard planar processing including a top annular ohmic contact with a 5μmdiam pinhole in the center to couple out enough of the internal emission for accurate freespace power measurements [4]. The emission spectra have the behavior displayed in Fig. 1(b), parameterized by bias voltage (VB). The long wavelength emission edge is at = 1684 nm  close to the In0.53Ga0.47As bandgap energy of Ug ≈ 0.75 eV at 300 K. The spectral peaks for VB = 2.8 and 3.0 V both occur around = 1550 nm (h = 0.75 eV), so blueshifted relative to the peak of the “ideal”, bulk InGaAs emission spectrum shown in Fig. 1(b) [5]. These results are consistent with the model displayed in Fig. 1(c), whereby the broad emission peak is attributed to the radiative recombination between electrons accumulated on the emitter side, and holes generated on the emitter side by interband tunneling with current density Jinter. The blueshifted main peak is attributed to the quantumsize effect on the emitter side, which creates a radiative recombination rate RN,2 comparable to the bandedge crossgap rate RN,1. Further support for this model is provided by the shorter wavelength and weaker emission peak shown in Fig. 1(b) around = 1148 nm. Our quantum mechanical calculations attribute this to radiative recombination RR,3 in the RTD quantum well between the electron groundstate level E1,e, and the hole level E1,h. To further test the model and estimate quantum efficiencies, we conducted optical power measurements using a largearea Ge photodiode located ≈3 mm away from the RTD pinhole, and having spectral response between 800 and 1800 nm with a peak responsivity of ≈0.85 A/W at =1550 nm. Simultaneous IV and LV plots were obtained and are plotted in Fig. 2(a) with positive bias on the top contact (emitter on the bottom). The IV curve displays a pronounced NDR region having a current peaktovalley current ratio of 10.7 (typical for In0.53Ga0.47As RTDs). The external quantum efficiency (EQE) was calculated from EQE = e∙IP/(∙IE∙h) where IP is the photodiode dc current and IE the RTD current. The plot of EQE is shown in Fig. 2(b) where we see a very rapid rise with VB, but a maximum value (at VB= 3.0 V) of only ≈2×105. To extract the internal quantum efficiency (IQE), we use the expression EQE= c ∙i ∙r ≡ c∙IQE where ci, and r are the opticalcoupling, electricalinjection, and radiative recombination efficiencies, respectively [6]. Our separate optical calculations yield c≈3.4×104 (limited primarily by the small pinhole) from which we obtain the curve of IQE plotted in Fig. 2(b) (righthand scale). The maximum value of IQE (again at VB = 3.0 V) is 6.0%. 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And because of the highquality of the InGaAs/AlAs heterostructure (very few traps or deep levels), most of the holes should reach the emitter side by some combination of drift, diffusion, and tunneling through the valenceband double barriers (TypeI offset) between InGaAs and AlAs. The computed interband current density Jinter is shown in Fig. 3(a) along with the total current density Jtot. At the maximum Jinter (at VB=3.0 V) of 7.4×102 A/cm2, we get i = Jinter/Jtot = 0.18, which is surprisingly high considering there is no ptype doping in the device. When combined with the Augerlimited r of 0.41 and c ≈ 3.4×104, we find a model value of IQE = 7.4% in good agreement with experiment. This leads to the model values for EQE plotted in Fig. 2(b)  also in good agreement with experiment. Finally, we address the high Jinter and consider a possible universal nature of the lightemission mechanism. Fig. 3(b) shows the tunneling probability T according to the Kane twoband model in the three materials, In0.53Ga0.47As, GaAs, and GaN, following our observation of a similar electroluminescence mechanism in GaN/AlN RTDs (due to strong polarization field of wurtzite structures) [8]. The expression is Tinter = (2/9)∙exp[(2 ∙Ug 2 ∙me)/(2h∙P∙E)], where Ug is the bandgap energy, P is the valencetoconductionband momentum matrix element, and E is the electric field. Values for the highest calculated internal E fields for the InGaAs and GaN are also shown, indicating that Tinter in those structures approaches values of ~105. As shown, a GaAs RTD would require an internal field of ~6×105 V/cm, which is rarely realized in standard GaAs RTDs, perhaps explaining why there have been few if any reports of roomtemperature electroluminescence in the GaAs devices. [1] E.R. Brown,et al., Appl. Phys. Lett., vol. 58, 2291, 1991. [5] S. Sze, Physics of Semiconductor Devices, 2nd Ed. 12.2.1 (Wiley, 1981). [2] M. Feiginov et al., Appl. Phys. Lett., 99, 233506, 2011. [6] L. Coldren, Diode Lasers and Photonic Integrated Circuits, (Wiley, 1995). [3] Y. Nishida et al., Nature Sci. Reports, 9, 18125, 2019. [7] E.O. Kane, J. of Appl. Phy 32, 83 (1961). [4] P. Fakhimi, et al., 2019 DRC Conference Digest. [8] T. Growden, et al., Nature Light: Science & Applications 7, 17150 (2018). [5] S. Sze, Physics of Semiconductor Devices, 2nd Ed. 12.2.1 (Wiley, 1981). [6] L. Coldren, Diode Lasers and Photonic Integrated Circuits, (Wiley, 1995). [7] E.O. Kane, J. of Appl. Phy 32, 83 (1961). [8] T. Growden, et al., Nature Light: Science & Applications 7, 17150 (2018).more » « less

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