Joule Heating and Phonon Transport in SiliconMOSFETs

Z. Aksamija, and U. RavaioliBeckman Institute, University of Illinois, 405 N. Mathews, 61801 Urbana IL., U.S.A.

e-mail: aksamija@uiuc.edu

INTRODUCTION

Joule heating is caused by emission of phononsas electrons traverse through a semiconductor de-vice. In silicon MOSFETs, most of the emission isconcentrated in the small region where the channelmeets the drain. It has been noted in the literaturethat this causes a hot-spot in the device where strongnon-equilibrium conditions exist [1]. The emissionof phonons has previously been examined, but theresulting non-equilibrium temperature conditions ina typical device have never been established. Thiswork aims at using full-band Monte Carlo simula-tion coupled with phonon transport to quantify theextent and the location of Joule heating in a siliconMOSFET.

NUMERICAL APPROACHES

Data on scattering events in a device can beobtained from a Monte Carlo simulation. In thiswork we use a three-dimensional full-band simula-tor with a self-consistent non-linear Poisson solver.In order to insure accuracy of results for phononevents, a full phonon dispersion is included. It iscalculated from an Adiabatic Bond Charge modeland tabulated for lookup [2]. An iterative algorithmwas devised in order to make all scattering eventsinvolving phonons energy and momentum conserv-ing with the full phonon dispersion relationship.This gives us a more accurate value for the phononmomentum and energy.

Once the simulation run is complete, data onall phonon events that occurred is tabulated. Thenphonon velocity is looked up from the dispersionrelationship for each phonon, and each phonon isallowed to move without scattering until the endof the simulation time-frame. Finally, the entiresimulation region is divided into small cubes, andthe total phonon energy in each cube is computed.

A curve relating lattice energyU(T ) and tempera-ture T is pre-computed by integrating the phonondensity-of-states (DOS)D(ω) with the phonon oc-cupation probability〈n(ω)〉, as given by (1).

U(T ) =∫

dωD(ω)〈n(ω)〉h̄ω (1)

The phonon DOS is calculated numericallyfrom tabulated dispersion data using the algorithmproposed in [3]. Using these results, the non-equilibrium temperature in each location of thedevice can be found.

RESULTS

The computed phonon DOS is shown in Fig. 1. Itwas found that transitions between opposite equiv-alent valleys dominate the scattering statistics, asshown in Fig. 2. The reduced momenta of thephonons resulting from such transitions are nearthe Brillouin zone edge and rest on the longitudinaloptical branch. The direction of propagation of thesephonons is perpendicular to the energy isosurface,shown in Fig. 3, so most of the phonons emittedcontinue to travel in the direction of the flow ofelectrons. This means very little heat actually flowstowards the substrate, as in Fig. 4. Future work isplanned to enable coupled electro-thermal simula-tion in order to probe this process in more detail.

REFERENCES

[1] S. Sinha, P. K. Shelling, S. R. Phillpot, and K. E. Goodson,Scattering of g-process longitudinal optical phonons athotspots in silicon, Journal of Applied Physics97, 023702(2005).

[2] W. Weber,Adiabatic bond charge model for the phononsin diamond, Si, Ge andα-Sn, Physical Review B15, 10(1977).

[3] G. Gilat and L. J. Raubenheimer,Accurate NumericalMethod for Calculating Frequency-Distribution Functionsin Solids, Physical Review144, 2 (1966).

Fig. 1. Phonon DOS computed numerically. The dashed lineis the standard Debye approximation, for comparison.

Fig. 2. Number of phonons emitted for each event type.Opposite equivalent valley and acoustic intravalley transitionsare marked. Opposite equivalent valley transitions dominate.

Fig. 3. Surface of equal phonon energy in momentumspace for the longitudinal optical branch. Direction of phononpropagation is perpendicular to this surface.

Fig. 4. Map of temperature distribution in an example shortchannel MOSFET device after 20ps. The clouds of elevatedtemperature have moved to the left of the source and drainregions.