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RESEARCH PAPERS: Micro/Nanoscale Heat Transfer

# Non-Equilibrium Phonon Distributions in Sub-$100nm$ Silicon Transistors

[+] Author and Article Information
S. Sinha

Thermosciences Division, Mechanical Engineering Department,  Stanford University, California 94305-3030sanjiv@stanfordalumni.org

E. Pop, R. W. Dutton

Electrical Engineering Department,  Stanford University, California 94305-3030

K. E. Goodson

Thermosciences Division, Mechanical Engineering Department,  Stanford University, California 94305-3030

J. Heat Transfer 128(7), 638-647 (Dec 21, 2005) (10 pages) doi:10.1115/1.2194041 History: Received June 07, 2005; Revised December 21, 2005

## Abstract

Intense electron-phonon scattering near the peak electric field in a semiconductor device results in nanometer-scale phonon hotspots. Past studies have argued that ballistic phonon transport near such hotspots serves to restrict heat conduction. We reexamine this assertion by developing a new phonon transport model. In a departure from previous studies, we treat isotropic dispersion in all phonon branches and include a phonon emission spectrum from independent Monte Carlo simulations of electron-phonon scattering. We cast the model in terms of a non-equilibrium phonon distribution function and compare predictions from this model with data for ballistic transport in silicon. The solution to the steady-state transport equations for bulk silicon transistors shows that energy stagnation at the hotspot results in an excess equivalent temperature rise of about 13% in a $90nm$ gate-length device. Longitudinal optical phonons with non-zero group velocities dominate transport. We find that the resistance associated with ballistic transport does not overwhelm that from the package unless the peak power density approaches $50W∕μm3$. A transient calculation shows negligible phonon accumulation and retardation between successive logic states. This work highlights and reduces the knowledge gaps in the electro-thermal simulation of transistors.

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## Figures

Figure 1

Contours of heat generation in a 90nm gate-length bulk silicon n-MOSFET are calculated using the hydrodynamic model for electron transport. The peak power density at the center is nearly 5W∕μm3.

Figure 2

A cross section of the experimental structure used to probe ballistic conduction near a doped resistor in silicon (14) is shown at the top. The resistor acted as a hotspot inside the silicon membrane. The symmetry in the problem is used to solve the BTE in the domain shown below.

Figure 3

A comparison of the thermal resistance measured at different temperatures. Predictions based on the proposed model are given by the continuous line. The predictions suffer from lack of information on temperature dependent scattering rates of phonons emitted by holes in silicon.

Figure 4

The boundary conditions used in the device calculations are shown above. Heat is assumed to flow out through the metallic contacts at the top which act as fins. The heat transfer coefficient is relatively high due to this spreading effect. Most of the heat flow is toward the heat sink through the bulk silicon at the bottom.

Figure 5

The cylindrical geometry resulting from the assumption that the hotspot is located at the center of the channel serves to reduce the complexity. We further assume a step profile with the heat source confined to a radius a, as shown schematically.

Figure 6

The phonon dispersion in silicon along [100] is obtained from a fit to neutron scattering data from Dolling (23). The above dispersion is assumed to hold along all directions in our calculations. At an electric field of 4MV∕m, the source term in the BTE, ṅk,s, has the frequency dependence shown on the right.

Figure 7

The phonon number density for different branches is shown close to the hotspot, which is 20nm in extent. The LO contribution dominates the number density and consequently, the energy density.

Figure 8

The non-Fourier heat flux due to different phonon branches is shown. Ballistic heat conduction is predominantly through LO phonons. The cumulative flux is only about 5% of the total, the rest being the flux due to thermalized phonons that obey the Fourier law.

Figure 9

A comparison of the temperature fields obtained from continuum heat diffusion and phonon heat conduction is shown above. Ballistic conduction augments the overall thermal resistance between the transistor and the ambient by about 13%.

Figure 10

The distribution of free paths in room temperature silicon as a function of phonon frequency and polarization is shown. The mean free path of phonons emitted by hot electrons in a device are also given for comparison. The use of a gray-body approximation for the heat source would lead to significant errors in predictiing the ballistic nature of transport near hotspots.

Figure 12

Contours of the normalized phonon number density, spaced by 0.01, are shown as a function of position and time for the longitudinal optical phonon at 14THz. No wave retardation is evident since the emitted phonons have large enough group velocities. The accumulation of LO phonons near the source during the time period of power dissipation is clearly visible.

Figure 13

Contours of the normalized phonon number density, spaced by 0.01, are shown as a function of frequency and time at x=0. There is no phonon accumulation for a switching period of 100ps with a duty cycle of 30%.

Figure 11

A step-like phonon source symmetric about x=0 with a uniform power density of 5W∕μm3 is considered for a sample transient calculation. The extent of the source, a, is taken to be 20nm, consistent with device hotspots. A sink at 300K is assumed to be present at x=±l where l=300nm.

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