Joule Heating and Phonon Transport in Silicon...
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Joule Heating and Phonon Transport in Silicon MOSFETs Z. Aksamija, and U. Ravaioli Beckman Institute, University of Illinois, 405 N. Mathews, 61801 Urbana IL., U.S.A. e-mail: [email protected] I NTRODUCTION Joule heating is caused by emission of phonons as electrons traverse through a semiconductor de- vice. In silicon MOSFETs, most of the emission is concentrated in the small region where the channel meets the drain. It has been noted in the literature that this causes a hot-spot in the device where strong non-equilibrium conditions exist [1]. The emission of phonons has previously been examined, but the resulting non-equilibrium temperature conditions in a typical device have never been established. This work aims at using full-band Monte Carlo simula- tion coupled with phonon transport to quantify the extent and the location of Joule heating in a silicon MOSFET. NUMERICAL APPROACHES Data on scattering events in a device can be obtained from a Monte Carlo simulation. In this work 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 phonon events, a full phonon dispersion is included. It is calculated from an Adiabatic Bond Charge model and tabulated for lookup [2]. An iterative algorithm was devised in order to make all scattering events involving phonons energy and momentum conserv- ing with the full phonon dispersion relationship. This gives us a more accurate value for the phonon momentum and energy. Once the simulation run is complete, data on all phonon events that occurred is tabulated. Then phonon velocity is looked up from the dispersion relationship for each phonon, and each phonon is allowed to move without scattering until the end of the simulation time-frame. Finally, the entire simulation region is divided into small cubes, and the total phonon energy in each cube is computed. A curve relating lattice energy U (T ) and tempera- ture T is pre-computed by integrating the phonon density-of-states (DOS) D(ω) with the phonon oc- cupation probability n(ω), as given by (1). U (T )= dωD(ω)n(ω)¯ hω (1) The phonon DOS is calculated numerically from tabulated dispersion data using the algorithm proposed in [3]. Using these results, the non- equilibrium temperature in each location of the device can be found. RESULTS The computed phonon DOS is shown in Fig. 1. It was found that transitions between opposite equiv- alent valleys dominate the scattering statistics, as shown in Fig. 2. The reduced momenta of the phonons resulting from such transitions are near the Brillouin zone edge and rest on the longitudinal optical branch. The direction of propagation of these phonons is perpendicular to the energy isosurface, shown in Fig. 3, so most of the phonons emitted continue to travel in the direction of the flow of electrons. This means very little heat actually flows towards the substrate, as in Fig. 4. Future work is planned 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 at hotspots in silicon, Journal of Applied Physics 97, 023702 (2005). [2] W. Weber, Adiabatic bond charge model for the phonons in diamond, Si, Ge and α-Sn, Physical Review B 15, 10 (1977). [3] G. Gilat and L. J. Raubenheimer, Accurate Numerical Method for Calculating Frequency-Distribution Functions in Solids, Physical Review 144, 2 (1966).
Transcript of Joule Heating and Phonon Transport in Silicon...
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: [email protected]
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.
