Joule Heating in Monolayer Graphene Field-effect Transistors

YOUNGWOOD, Pa. - April 17, 2013 - PRLog -- [Background

Graphene comprised of 2-D sheets of single carbon atom monolayers is of great interest in FETs. FET stands for Field-effect transistors. Typically, the FET monolayer [1] is about 2.65 microns long x 1.45 microns wide supported on a 300 nm SiO₂ substrate backed by a thick layer of silicon, the cross-section of which taken through the length of the FET is shown in the thumbnail. Bias voltages are applied through Pd/Au contact layers formed along the full width of the monolayer edge. The FET supports electrical power densities to 210 kW/cm².

Currently, the heat transfer analysis [1] of Joule heat in FETs is based on the Fourier diffusion equation, e.g., the PDEase finite element program, the temperature contours of which are depicted in the right-side of the thumbnail. Classically, the graphene may lose heat by EM thermal BB radiation. EM stands for electromagnetic and BB for blackbody. However, BB radiation from the graphene at the peak temperature of 1000 K is negligible. Even at FET temperatures [2] of 2000 K, the BB radiated power remains an insignificant fraction of the Joule heat.  Hence, the Joule heat may only be balanced by thermal conduction lateral to the graphene into the SiO₂ substrate and underlying silicon layer.

Problem

Unlike classical physics, QM requires the graphene to emit QED induced non-thermal EM radiation as noted in the thumbnail.  QM stands for quantum mechanics and QED for quantum electrodynamics.  Finding basis in the QM requirement that the heat capacity of the atom vanishes at the nanoscale, Joule heat generated in FETs cannot be conserved by an increase in temperature. Instead, conservation proceeds by the QED induced frequency up-conversion of the Joule heat within the FET to non-thermal EM radiation at the TIR confinement frequency of the graphene monolayer. TIR stands for total internal reflection.

The problem is the QED radiation loss required by QM is not included in the classical heat balance of the graphene, the omission [1, 2] leading to the conclusion that Joule heat produces temperatures from 1000 to 2000 K. However, if QED radiation is included in the heat balance, Joule heat does not increase the graphene temperature. See QED induced radiation at http://www.nanoqed.org at 2009 – 2013.

Discussions

Validity of TIR Confinement  The only requirement for TIR confinement is that the refractive index n of the graphene is greater than that of the surroundings.  In FETs [1, 2], the requirement is satisfied as graphene has n = 2.69 while that of the air or vacuum above the graphene and the SiO₂ below the graphene have n = 1 and 1.544, respectively.

Planck Energy of QED Radiation TIR confinement of atomically thin graphene induces the creation of QED radiation having very high Planck energy. In graphene, QED radiation is induced at a TIR wavelength L = 2nd, where d is the atomic layer thickness.  For d < 1 nm, L < 5 nm corresponding to Planck energies > 250 eV. Hence, QED radiation at soft X-ray levels is produced that not only passes through the space above the graphene, but also through the SiO₂ and silicon layers below the graphene. Although temperatures [1, 2] are not reported for the silicon layer, any temperature increase requires the unlikely absorption of soft X-rays in silicon having thicknesses of a few microns. Therefore, QM predicts the temperature does not increase in the silicon layer.  

Temperatures inferred from Raman shift Currently, spatially resolved Raman spectroscopy based on the G and 2D peaks is used as a microscopic thermometer to measure [1-3] the local temperature distribution in graphene FETs. However, the notion that effective temperatures may be inferred for a particular vibration mode has a long history [4] of controversy where it was shown laser excitation of molecules is not equivalent to thermal heating, and therefore the effective temperature of a laser excited Raman vibration mode of a molecule has no meaning.

The problem with assigning an effective temperature to a particular vibration mode of a molecule is an artifact of statistical mechanics. Statistical mechanics is applicable to the effective vibrational temperature of a very large ensemble of molecules, but not to the temperature of the molecule itself. In contrast, QM precludes the single molecule from having the heat capacity to conserve the absorption of EM energy by an increase in temperature, thereby requiring conservation to proceed by the molecule dissipating the absorbed EM energy by vibration that is transferred from molecule to molecule and eventually absorbed by the container walls. By QM, the macroscopic container walls increase in temperature, but not the molecules themselves. Hence, the effective temperature of a particular vibration mode of a single molecule based on statistical mechanics has no meaning, and therefore the effective temperature of graphene FETs inferred from Raman shifts measured during Joule heating is highly questionable. In fact, the experimental correlation [5] between Raman shifts and the controlled temperature of graphene monolayer specimens is meaningless, even if one sets aside the fact that graphene specimens at a controlled temperature are most likely not at the same temperature as those in FETs under Joule heating.
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Conclusions

1. The QED induced non-thermal EM radiation created in FETs because of QM  precludes the graphene atoms from having the heat capacity to conserve Joule heat by an increase in temperature thereby invalidating inferred FET temperatures by Raman shift  measurements of G and 2D peaks. Similarly. QM refutes temperatures in nanoscale junctions inferred by the ratio of anti-Stokes to Stokes Raman shifts.

2. Measurement of the silicon layer temperature is expected to show Joule heat does not increase FET temperatures.  

3. The QED radiation in atomic graphene monolayers has Planck energy at soft X-ray levels that passes through most materials and would not be absorbed in the silicon substrate to allow an increase in temperature to be detected.

References

[1] M. Freitag, et al., "Energy Dissipation in Graphene Field-Effect Transistors," Nano Lett., vol. 9, pp. 1883 – 1888, 2009.
[2] S. Berciaud, et al., “Electron and Optical Phonon Temperatures in Electrically Biased Graphene,” PRL, 104, pp. 227401, 2010.
[3] M. Oron-Carl and R. Krupke, “Raman Spectroscopic Evidence for Hot-Phonon Generation in Electrically Biased Carbon Nanotubes,” PRL, vol. 100, pp. 127401, 2008.
[4] E. R. Grant, et al., “Is Multiphoton Dissociation of Molecules a Statistical Thermal Process?” PRL, vol. 40, pp. 115 - 118, 1978.
[5] I. Calizo, et al., “Variable Temperature Raman Spectroscopy as a nanometrology tool for graphene layers and graphene-based devices,” Appl. Phys. Lett., vol. 91, pp. 071913, 2007.