Recap (so far) - Stanford University

© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips 1

Recap (so far)

• Ohm’s & Fourier’s Laws

• Mobility & Thermal Conductivity

• Heat Capacity

• Wiedemann-Franz Relationship

• Size Effects and Breakdown of Classical Laws


Low-Dimensional &

Boundary Effects

• Energy Transport in Thin Films, Nanowires, Nanotubes

• Landauer Transport

− Quantum of Electrical and Thermal Conductance

• Electrical and Thermal Contacts

• Materials Thermometry

• Guest Lecture: Prof. David Cahill (MSE)

© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips

L ~ 200 nm

Si

D

Si

Ox

• Size and Non-Equilibrium Effects

− optical-acoustic

− small heat source

− impurity scattering

− boundary scattering

− boundary resistance

• Macroscale (D >> L)

• Nanoscale (D < L)

QTkt

TC ss

Qee

evt

e

phon

eq

“Sub-Continuum” Energy Transport

Ox Me

tsi


Thermal Simulation Hierarchy

4

defect

lattice wave

phononE

L

D

D ~ L

Waves & Atoms

Continuum

Fourier’s Law, FE

Phonon Transport

BTE & Monte Carlo

Waves & Atoms

MD & QMD

D ~

MFP ~ 200 nm at 300 K in Si

q

qq

qq

q nnnv

t

n

.

Tkq

"

Wavelength


Thermal and Electrical Simulation

5

AtomisticP

ho

no

ns

Diffusion

BTE or

Monte Carlo

BTE with

Wave models

Dri

ft D

iffu

sio

n

BT

E

Mo

men

ts

Mo

nte

Carl

o

& B

TE

Mo

nte

Carl

o

wit

h Q

uan

tum

Mo

dels

Electrons

Fu

ll Q

uan

tum

Isothermal

~1 nm~5 nm

~100 nm~5 nmMFP

phononselectrons


Nanowire Formation: “Bottom-Up”

• Vapor-Liquid-Solid (VLS) growth

• Need gas reactant as Si source

(e.g. silane, SiH4)

• Generated through

– Chemical vapor deposition (CVD)

– Laser ablation or MBE (solid target)

6

Lu & Lieber, J. Phys. D (2006)


• “Top-down” = through

conventional lithography

• “Guided” growth = through porous

templates (anodic Al2O3)

– Vapor or electrochemical

deposition

7

Suspended nanowire (Tilke „03)

“Top-Down” and Templated Nanowires


Semimetal-Semiconductor Transition

• Bi (bismuth) has

semimetal-semiconductor

transition at wire D ~ 50 nm

due to quantum

confinement effects

8

Source: M. Dresselhaus (MIT)


When to Worry About Confinement

9

d

2-D Electrons 2-D Phonons

2

2 2

n n y z

nvk v k k

d

22

*2n

nE

m d

d


Nanowire Applications

• Transistors

• Interconnects

• Thermoelectrics

• Heterostructures

• Single-electron devices

10


Nanowire Thermal Conductivity

11

Li, Appl. Phys. Lett. 83, 3187 (2003)

Nanowire diameter


Interconnects = Top-Down Nanowires

12

SEM of AMD‟s “Hammer” microprocessor in 130 nm CMOS with 9 copper layers

Intel 65 nm

Cross-section8 metal levels + ILD

TransistorM1 pitch


Cu Resistivity Increase <100 nm Lines

• Size Matters

• Why?

• Remember

Matthiessen’s Rule

13


Cu Interconnect Delays Increase Too

Source: ITRS http://www.itrs.net

14

http://www.itrs.net/


Industry Acknowledged Challenges

15

Source: ITRS http://www.itrs.net


Cu Resistivity and Line Width

16

Steinhögl et al., Phys. Rev. B66 (2002)

http://www.itrs.net/


Modeling Cu Line Resistivity

17



Model Applications

18


Plombon et al., Appl. Phys. Lett 89 (2006)


Resistivity Temperature Dependence

19


Other Material Resistivity and MFP

• Greater MFP (λ) means greater impact of “size effects”

• Will Aluminum get a second chance?!

20


Same Effect for Thermal Conductivity!

• Material with longer (bulk, phonon-limited) MFP λ

suffers a stronger % decrease in conductivity in thin films

or nanowires (when d ≤ λ)

• Nanowire (NW) data by Li (2003), model Pop (2004)

21

0

10

20

30

40

50

60

70

80

0 50 100 150d (nm)

k (

W/m

/K) Thin Si

SiGe NW

Si NW

Thin Ge

Recall:

• bulk Si kth ~ 150 W/m/K

• bulk Ge kth ~ 60 W/m/K

Approximate bulk MFP‟s:

• λSi ~ 100 nm

• λGe ~ 60 nm

(at room temperature)


Back-of-Envelope Estimates

22

0

10

20

30

40

50

60

70

80

0 50 100 150d (nm)

k (

W/m

/K) Thin Si

SiGe NW

Si NW

Thin Ge

1( )

3k d Cv

C

(MJm-3K-1)

λb

(nm)

vL

(m/s)

vT

(m/s)kb

(Wm-1K-1)

Si 1.66 ~100 9000 5330 150

Ge 1.73 ~60 5000 3550 60

1 1 1 1

b Gd D

(at room temperature)


More Sophisticated Analytic Models

23

δ = d/λ < 1 S = (1 – δ2)1/2

Flik and Tien, J. Heat Transfer (1990) Goodson, Annu. Rev. Mater. Sci. (1999)


A Few Other Scenarios

24

Goodson, Annu. Rev. Mater. Sci. (1999)

anisotropy


Onto Nanotubes…

• Nanowires:

– “Shrunk-down” 3D cylinders of a larger solid (large surface area

to volume ratio)

– Diameter d typically < {electron, phonon} bulk MFP Λ: surface

roughness and grain boundary scattering important

– Quantum confinement does not play a role unless d < {electron,

phonon} wavelength λ ~ 1-5 nm (rarely!)

• Nanotubes:

– “Rolled-up” sheets of a 2D atomic plane

– There is “no” volume, everything is a surface*

– Diameter 1-3 nm (single-wall) comparable to wavelength λ so

nanotubes do have 1D characteristics

25

* people usually define “thickness” b ~ 0.34 nm

b


Single-Wall Carbon Nanotubes

26

• Carbon nanotube = rolled up graphene sheet

• Great electrical properties

– Semiconducting Transistors

– Metallic Interconnects

– Electrical Conductivity σ ≈ 100 x σCu

– Thermal Conductivity k ≈ kdiamond ≈ 5 x kCu

HfO2

S (Pd) D (Pd)

SiO2

top gate (Al) CNT

d ~ 1-3 nm

• Nanotube challenges:

– Reproducible growth

– Control of electrical and thermal properties

– Going “from one to a billion”


CVD Growth at ~900 oC

27


Fe Nanoparticle-Assisted Nanotube Growth

• Particle size corresponds to nanotube diameter

• Catalytic particles (“active end”) remain stuck to substrate

• The other end is dome-closed

• Base growth

28


Water-Assisted CVD and Breakdown

• People can also grow

“macroscopic” nanotube-

based structures

• Nanotubes break down at

~600 oC in O2, 1000 oC in N2

29

Hata et al., Science (2004)

in N2

in O2


Graphite Electronic Structure

30

b ~ 3.4 Å

aCC ~ 1.42 Å

|a1| = |a2| = √3aCC

http://www.photon.t.u-tokyo.ac.jp/~maruyama/kataura/discussions.html





Nanotube Electronic Structure

31

EG > 0

EG = 0

EG > 0

EG = 0

Collin

s a

nd A

vouris,

Scie

ntific A

merican

(2000)


Band Gap Variation with Diameter

32

• Red: metallic

• Black: semiconducting

http://www.photon.t.u-tokyo.ac.jp/~maruyama/kataura/kataura.html

E11,M

E11,M

E22,M

E22,S

E11,S = EG

≈ 0.8/d

Charlier, Rev. Mod. Phys. (2007)

“Kataura plot”





Nanotube Current Density ~ 109 A/cm2

• Nanotubes are nearly

ballistic conductors up to

room temperature

• Electron mean free path ~

100-1000 nm

33

S (Pd) D (Pd)

SiO2

CNT

G (Si)

Javey et al., Phys. Rev. Lett. (2004)

L = 60 nmVDS = 1 mV


Transport in Suspended NanotubesE. Pop et al., Phys. Rev. Lett. 95, 155505 (2005)

SiO2

Si3N4

nanotube Pt

Pt gate

2 μmnanotube on

substrate suspended

over trench

• Observation: significant current degradation and negative

differential conductance at high bias in suspended tubes

• Question: Why? Answer: Tube gets HOT (how?)


1

, ,

1 1 1eff

AC OP ems OP abs

Include OP absorption:

Transport Model Including Hot Phonons

),(

),(

4),(

2 TV

TVL

q

hRTVR

eff

eff

C

0( )OP AC ACT T T T

Non-equilibrium OP:

T0

TAC = TL

TOP

RTH

ROP

I2(R-Rc)

0 0.2 0.4 0.6 0.8 1 1.2

300

400

500

600

700

800

900

1000

V (V)

Ph

on

on

Te

mp

era

ture

(K

)

oxidation T

Optical TOP

Acoustic TAC

I2(R-RC)

TOP

TAC = TL

2( ) ( ) / 0CA k T I R R L

Heat transfer via AC:

Landauer electrical resistance

E. Pop et al., Phys. Rev. Lett. 95, 155505 (2005)


Extracting SWNT Thermal Conductivity

• Ask the “inverse” question: Can I extract thermal properties from electrical data?

• Numerical extraction of k from the high bias (V > 0.3 V) tail of I-V data

• Compare to data from 100-300 K of UT Austin group (C. Yu, NL Sep’05)

• Result: first “complete” picture of SWNT thermal conductivity from 100 – 800 K

E. Pop et al., Nano Letters 6, 96 (2006)

Yu et al. (NL‟05)This work

~T

~1/T

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