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Thermal Coupling

-Fundamental physics-Efects & Analysis in nanometric ICs-Papers

Outline

Physical principlesSources of power consumptionThermal coupling / Self heating modelsEffects of temperature on circuit performanceEffects of temperature on circuit reliabilityCooling strategiesTemperature monitoring

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Physical Principles: Definitions

Thermal EnergyKinetic and potential energy of particles. Phonons

HeatEnergy transferred between bodies at different Temperature

Relation between Energy-Temperature

E: Heat or Energy. T: Temperature. C: Thermal capacitance. c: specific heat.Similar to: Capacitance, charge, Voltage.WARNING: CHANGES OF PHASE!!1 cal = 4,184 J

TcmTCE ∆⋅⋅=∆⋅=

Specific Heat: Some examples

Substance c, KJ/Kg·K c, Kcal/Kg·K - Btu/lb·F°

H2O 4,18 1,00Ethylic Alcohol 2,4 0,58Al 0,900 0,215Cu 0,386 0,0923Au 0,126 0,0301Ag 0,233 0.0558Glass 0,840 0,20Ice ( -10 °C) 2,05 0,49Si 0.703 0.168

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Heat flow – Power – Heat transfer

Energy ratesJ/s = WTemporal Evolution of Temperature (Adiabatic condition)

Examples of heat generation:Joule EffectScattering of carriers in the conductor lattice and phonons

Temporal evolution of temperature (non adiabatic):Conservation of Energy

Stored energy (∆T) = Generated Energy – transferred energy

dtdTCtQtP

dtdE

⋅=== )()(

Modes of heat transfer: Phoenomenological laws

Heat conduction in solids: Fourier lawq: W/m2

k: thermal conductivityW/(K·m)

Q: W

Thermal resistancedxdTkq

AQTkq

−=

=∇−=

T1 T2

Q

AL

kxkxAdxRth

QRthTx

x

⋅==

=∆

∫1

)()·(

·2

1

4

Thermal conductivity. Examples

Sik(T) = 286/(T-100) W/cm · Kk = 1,05 W/cm·K (t = 100°C)

Cuk = 3,93 W/cm·K

Alumina (99-99,5 %)k = 0,25 W/cm·K

Epoxi (conductive)k = 0,016 W/cm·K

Epoxi (non conductive)k = 0,004 W/cm·K

Beriliak = 2,05 W/cm·K (t = 20 °C)

Aluminak = 0,17 W/cm·K (t = 20 °C)

Exercise: Thermal resistance calculation

Different shapesArea Sphere: 4πr2

Area cylinder: 2πrhCase 1 Case 2,3: Concentric cylinder (H), sphere

H

Ai=L2

Angle: 45º

r1

r2

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Convection heat transfer

DefinitionEnergy transferred to a moving fluid

Convection classificationNatural: Difference of densitiesForced: Fan or pump

Different classification of fluidsLaminar, turbulent…

Fluid, Tf,vf

qTs

Convection heat transfer: Newton cooling law

Phenomenological relationship

hc: Convective coefficient of heat transfer

Units: [W/m2·K]Depends on: fluid and convection types, viscosity, orientation, temperature differences…Air. Natural convection: 3-25Water. Natural convection : 15-100Air. Forced convection: 10-200Water. Forced convection: 50-10.000

( )fsc TThq −⋅=

fsConv

cConv

TTRQAh

R

−=⋅⋅

=1

Thermal resistance due to convection

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Convection Heat TransferRS 401-497100*64,5*154 °C/W

RS 401-403100*123,8*26,72,1 °C/W

RS 401-807152*130*321,1 °C/W

RS 401-958115*120*1200,5 °C/W

Radiation heat transfer: Black body and real surfaces

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51

−=

−

TCb eCq λλλ

C1 = 3,742·108 W·µm4/m2

C2 = 1,4389·104 µm·Kλ : µm.

Plank law

mKTmax µλ 6,2897· =

Wien law

4Tqb σ=σ=5.6697·10-8 W/m2 K4

Stefan-Bolzmann law

Each surface has its own emissivity εGray body: emissivity constant

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Heat transfer equation in an IC

Dissipating device:Heat source

Heat Sink (Infinite C)

Inside the IC: Conduction heat transferSilicon, Metal, Package

Convection, RadiationConduction (pins) to PCB

[1]

Heat Transfer Equation in an IC

Stored energy (DT) = Generated Energy – transferred energy

Expression per unit of volume:

Boundary conditions:- Convective (Realistic)- Isotermal- Adiabatic- Power dissipated as boundary condition

),(),(),(

),(),(),(

trTkrtqrttTc

trqrtqrttTc

GENERATED

CONDUCTEDGENERATED

∇∇+=∂∂

∇−=∂∂

ρ

ρ

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Sources of power consumption

Digital CMOS circuitsStatic power dissipation

Reverse-biased junction leakage currentGate direct tunneling leakageSubthreshold leakage (ISUB)

Dynamic power dissipationShort circuit currentCharging capacitances

MOS transistorRouting capacitances

[2]αfVCPDDLoadc25.0=

Sources of power consumption

MOS transistorsDissipation beneath the gate

Routing interconnectJoule EffectSurrounded by insulatorImportant heating as technology scales

Analogue CircuitsBiasSignal processing

[3]

[4]

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Sources of Power Consumtion: Evolution, density and distribution

[3]

[5]

[1]

Thermal coupling models

Problem definition

Procedure:Resolution of the heat transfer equation in the IC structureInitial and Boundary conditions

TechniquesAnalytical (Closed form or Fourier Decomposition)Circuital Model (Finite difference method)Numerical Solution

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Thermal coupling: Circuit model

Thermal coupling: characteristics at silicon level

Low pas filter behaviorCut off: 10 kHz – 1 MHz

Diffussion processVariable penetration depth (frequency dependent)Attenuation increasing with distance

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Thermal coupling: System level model

Thermal coupling examples:

Cross Section

Heat source1 2 3 4 N-2 N-1 N

Power Pulse

Dyn

amic

∆T Distance = 4 µm

Distance = 24 µm

Distance = 44 µm

Distance = 64 µm

∆T = 2°C

∆T = 0.2°C∆T = 0.1°C

∆T = 0.05°C

Magnitude = 13 mW

Time (µs)

0 10 20 30 40 50 60

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Effects of temperature on circuit performance

MOS transistor. Temperature affects:

Carrier mobility (MTE: 1.5-2)Threshold voltage (K: .5-4 mV/K)Velocity saturationDepletion layer depthSubthreshold current

WiresResistivity (ρ α T)

)()()(

)()( 00

rr

MTE

rr

TTKTVtTVtTTTT

−−=

= µµ

Effects of temperature on circuit performance

Effects on digital circuitsSignal Integrity

Clock skewDelayPower distribution

DelayStatic power consumption

Ring oscillator as temperature sensor

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Effects of temperature on circuit performance: Electrothermal simulation

Effects on circuit reliability

λ

λ

1at t urvivals

at t rateCasualty )(

0at tumber at t survivals ofNumber )(

=

=

==

MTBF

St

NtR Reliability of a component

Failure velocity

λ

TimeInfantil mortality

Util life

Old mortality

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Effects on circuit reliability

Temperature affects:Static working temperatureTemperature gradientsTemperature cyclingTemperature evolution

Arrhenius model

( )KTEe devref −⋅= λλLog λ

250 200 150 100Temperature

Cooling strategies

DissipatorsPassiveActive

Liquid coolingThermoelectric coolers

Peltier effect

A B

I

eABJq Π=

15

Temperature monitoring

Built-in temperature sensorsAbsoluteDifferential

Off chip techniquesContact techniques

Mechanical (AFM)Contrast materials: Phosphors, liquid crystal

Optical techniquesInfra-red emissionLaser based techniques

References

[1] Mahajan, R. et al. “Cooling a Microprocessor Chip”, Proceedings of IEEE, August 2006, Vol. 94, no. 8, pp. 1476-1486

[2] Pedram, M. et al. “Thermal Modeling, Analysis, and Management in VLSI Circuits: Principles and Methods”, Proceedings of IEEE, August 2006, Vol. 94, no. 8, pp. 1487-1501

[3] Lin S.C. et al. “Analysis and Implications of IC Cooling for Deep Nanometer Scale CMOS Technologies”, IEEE International Electron Device Meeting 2005. 2-7 Dec. 2005. pp. 1018-1021.

[4] Im, S. et al. “Scaling Analysis of Multilevel Interconnect Temperatures for High-Performance ICs”, IEEE Transactions on Electron Device, Vol. 52, no. 12, Dec. 2005.

[5] Gurrum, S.P., et al. “Thermal Issues in Next-Generation Integrated Circuits”, IEEE Transactions on Device and Materials Reliability, Vol. 4, no. 4, Dec. 2004. pp. 709-714