Strain analysis of nanostructures using Raman spectroscopy: theory and experiment

José MenéndezDepartment of PhysicsArizona State UniversityTempe, AZ 85287-USA

Just a second...

Mexico and us

30,000 Spanish refugees from Spanish War75,000 Americans moved to Mexico fleeing

McCarthysm50,000 Chileans fleeing dictator Pinochet

granted asylum200,000 Salvadorans fleeing civil war

granted Mexican residency225,000 Guatemalans fleeing counter-

insurgency granted refuge

Mexicoplatz in ViennaIn March of 1938 Mexico was the only country which officially protested at the League of Nations for the violent anexation of Austria to Nazi Germany.

In honor of this act the city of Vienna named this park Mexicoplatz.

Mexican heroes

Advisor: But shouldn’t we make a prior selection of refugees to be admitted into Mexico?

Cárdenas: Those who fought in their countries for their legal governments cannot be offended with an interrogation. We must take them all.

President Lázaro Cárdenas

Mexican heroes II

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Manuel Azaña, Spain’s last constitutional president, buried with the Mexican flag in France.

“For us it represents an honor, for the republicans, hope, and for you [the French government], a painful lesson”

Ambassador Luis Ignacio Rodríguez Taboada

Dealing with dictators

Mexico never recognized Spain’s illegitimate government. It restored diplomatic relations with Spain in 1977, two years after the dictator’s death.

Other countries had a different approach:

Semiconductor nanowires

•Dense packing

•FET performance close to balistic limit

•Wrap gates

•Core-shell: manipulation of optical and electronic properties via index mismatch, confinement, and strain.

J. Drucker

G. Liang et al., Nano Lett. 7, 642 (2007)

Acknowledgements

Rachna SinghEric DaileyPrashant MadrasJeff Drucker

“Seedless” nanowire growth

• Grown in CVD chamber on < 1ML Au / Si (111)

• Si wires (Si2H6) and Ge wires (Ge2H6)

•Tunable diameter and density

•<011> nanowire axis direction

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Eric Dailey and Jeff Drucker, J. App. Phys. 105, 064317 (2009)

Core-shell Ge-Si nanowires

Ge NWs were grown at 300 °C with 10 mTorr Ge2H6. Si shells were deposited at 540 °C, 3 mTorr Si2H6.

Critical thickness

I.A. Goldthorpe et al.,Nano Lett. v. 8, 4081 (2008)

Strain characterization with Raman spectroscopy

•Simple (No synchrotron needed)

•Non-destructive (unlike electron microscopy)

•Good spatial resolution.

Raman scattering

ωphonon(cm-1)= 2

λL−

kphonon2πn

4πn500 nm

π0.5 nm

Why is Raman affected by strain?

&&ui = ω0

2δ ik +K iklmε lm( )ukklm∑

3 phonon branches here!

Triple-degeneracy

Anharmonic coefficients. Same symmetry as elastic

tensor

Strain lifts the degeneracy...

pεxx +q ε yy + εzz( )−λ 2rεxy 2rεxz

2rεxy pε yy +q εxx + εzz( )−λ 2rε yz

2rεxz 2rε yz pεzz +q εxx + ε yy( )−λ

=0

λ = ω0

2 −ωstrained2( )

p = K11, q = K12, and r = K44 are anharmonic coefficients .

E. Anastassakis, A. Pinczuk, E. Burstein, F. Pollak, and M. Cardona, Sol. St. Comm. 8, 133 (1970).F. Cerdeira, C.J. Buchenauer, F.H. Pollak, and M. Cardona, Phys. Rev. B 5, 580 (1972)

Raman selection rules

Samplex

y

z

uy Ein/out Eout/in Ein/out

Eout/in Eout/in Ein/out

Phonon displacement

Light’s E-field

Light’s wave vector

Raman tensor

0 d 0

d 0 0

0 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

uz

0 0 0

0 0 d

0 d 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ux

0 0 d

0 0 0

d 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

uy

I i ∝ E incT ⋅R i( )⋅Escatt

2

Raman tensors

′R i( ) = L jiR j( )j

∑

L ji =uj ⋅ ′ui

Eigenvector of

perturbed phonon i

Eigenvector of un-

perturbed phonon j

I i ∝ E incT ⋅ ′R i( )⋅Escatt

2

Raman tensor of unperturbed

phonon j

Raman tensor of perturbed phonon

i

Can we get the strain tensor from Raman measurements?In principle possible. In practice

very hard.G. Loechelt et al., APL 66, 3639 (1995), JAP 86, 6164 (1999).

In nanostructures virtually impossible due to antenna effects.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

G. Chen et al Nano Lett. 85, 1341 (2008)

Basic equilibrium equations

∂σ xx

∂x+∂σ yx

∂y+∂σ zx

∂z+ fx =0

∂σ xy

∂x+∂σ yy

∂y+∂σ zy

∂z+ fy =0

∂σ xz

∂x+∂σ yz

∂y+∂σ zz

∂z+ fz =0

∂σ rr

∂r+σ rr −σθθ

r+1r∂σ rθ

∂θ+∂σ zr

∂z+ fr =0

∂σ rθ

∂r+2σ rθ

r+1r∂σθθ

∂θ+∂σ zθ

∂z+ fθ =0

∂σ rz

∂r+σ rz

r+1r∂σθz

∂θ+∂σ zz

∂z+ fz =0

Strain in cylindrical coordinates

ε ij =12

∂ui

∂xj

+∂uj

∂xi

⎛

⎝⎜

⎞

⎠⎟

εrr =∂ur

∂rεθθ =

1

r

∂ur

∂r+

ur

r

εrθ =1

r

∂ur

∂θ+

∂uθ

∂r−

uθ

r

ε zz =∂uz

∂z

εθz =1

r

∂uz

∂θ+

∂uθ

∂zεrz =

1

2

∂ur

∂z+

∂uz

∂r⎛⎝⎜

⎞⎠⎟

Cartesian

Cylindrical

Stress-strain relations in isotropic materials

σ rr =E

1+ν( ) 1−2ν( )1−ν( )εrr +ν εθθ +εzz( )⎡⎣ ⎤⎦

σθθ =E

1+ν( ) 1−2ν( )1−ν( )εθθ +ν εrr +εzz( )⎡⎣ ⎤⎦

σ zz =E

1+ν( ) 1−2ν( )1−ν( )εzz +ν εrr +εθθ( )⎡⎣ ⎤⎦

σθz =1+ν( )

Eεθz

σ rz =1+ν( )

Eεrz

σ rθ =1+ν( )

Eεrθ

The core-shell nanowire

c

a

GeSi

urr,θ,z( ) ≡ur r( ) uθ r,θ,z( ) =0 uz

r,θ,z( ) ≡uz z( )

Strain in core-shell nanowires

εrr =∂ur

∂r; εθθ =

ur

r; εzz =

∂uz

∂z

εrθ =0; εzr =0; εθz =0

⇒ σ rθ =0; σ zr =0; σθz =0

Equilibrium equations

∂σ rr

∂r+σ rr −σθθ

r=0

1r∂σθθ

∂θ=0

∂σ zz

∂z=0

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

d 2ur

dr 2+1r

dur

dr−

ur

r=0

d2uz

dz2=0

⎧

⎨⎪⎪

⎩⎪⎪

Boundary conditions

σ rrcore c( ) =σ rr

shell c( )

σ rrshell a( ) =0

σ zzshell a2 −c2( ) +σ zz

corec2 =0

urcore c( )−ur

shell c( ) =cεmisfit

uzcore l( )−uz

shell l( ) =lεmisfit

Same stress at interface

No stress @

outer surface

No net cross-section force

Shrink-fit

Shrink-fit

Pressing rings around shafts

Strain solution-core

εrrcore = −ε misfit a2 − c2

( )ν +1( ) 1− 2ν( )

1 − γ( ) 1− 2ν( )c2 − 1− 2ν + γ( ) a2 +ν

a2 − c2( ) + γ c2

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

ε zzcore = ε misfit

a2 − c2( )

a2 − c2( ) + γ c2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

εθθcore = ε rr

core

γ =E core

E shell

Strain solution-shell

εrrshell =

ε misfit ν +1( )γ c2

1 − γ( ) 1 − 2ν( )c2 − 1 − 2ν + γ( ) a2 1− 2ν( ) −a2

r2

⎡

⎣⎢

⎤

⎦⎥+

νγ c2ε misfit

a2 − c2( ) + γ c2

ε zzshell r,θ , z( ) = −

γ c2ε misfit

a2 − c2( ) + γ c2

γ =E core

E shell

εθθshell =

ε misfit ν +1( )γ c2

1 − γ( ) 1 − 2ν( )c2 − 1 − 2ν + γ( ) a2 1− 2ν( ) +a2

r2

⎡

⎣⎢

⎤

⎦⎥+

νγ c2ε misfit

a2 − c2( ) + γ c2

Core-shell strain

Optical phonons with strain

pε11 +q ε22 + ε33( )−λ 2rε12 2rε132rε12 pε22 +q ε11 + ε33( )−λ 2rε23

2rε13 2rε23 pε33 +q ε11 + ε22( )−λ=0

λ =ω 2 − ω02

Phonons are the solution of the eigenvalue problem:

1,2,3 are the crystal’s cubic cartesian

coordinates

<011> oriented nanowires

p +q( )εrr +qεzz −λ 0 00 1

2 p+ 32 q( )εrr + 1

2 p+q( )εzz −λ 2r εzz −εrr( )0 2r εzz −εrr( ) 1

2 p+ 32 q( )εrr + 1

2 p+q( )εzz −λ=0

2 x 2 problem can be easily diagonalized:

<111> oriented nanowires

13 p +2q( ) 2εrr +εzz( )−λ 2

3 r εzz −εrr( ) 23 r εzz −εrr( )

23 r εzz −εrr( ) 1

3 p+2q( ) 2εrr +εzz( )−λ 23 r εzz −εrr( )

23 r εzz −εrr( ) 2

3 r εzz −εrr( ) 13 p+2q( ) 2εrr +εzz( )−λ

=0

3 x 3 problem can be easily diagonalized:

Predictions for Ge-Si <011>

Predictions for Ge-Si <111>

Experiment: <011> nanowires

Eric Dailey and Jeff Drucker

Raman from pure Ge <011> nanowires

• Diameters 10-30 nm.

• Twice the Lorentzian linewidth as bulk Ge

• 1.2-1.4 cm-1 downshift. Not heating!

Raman from <011> Ge-Si core-shell nanowires

11 nm Ge / 5 nm Si

11 nm Ge / 3.4 nm Si

45 nm Ge / 9.1 nm Si

44 nm Ge / 3,7 nm Si

Core spectrum interpretationBroader, downshifted peak is Ge-

Ge mode from Si-Ge interface.Narrow, upshifted peak is from

Ge-core.Since no mode splitting obvious,

peak is assigned to LL mode.

Theoretical vs. experimental upshifts

Sample Exp. Theo. %

44/3.7 +2.2 cm-1

+4.37 cm-1 49.9

11/3.4 +3.1 +5.13 60.7

45/9.1 +1.8 +7.78 23.5

11/5 +7.7 +11.1 69.6

Critical thickness

I.A. Goldthorpe et al.,Nano Lett. v. 8, 4081 (2008)

Conclusions core Raman spectrumPredicted shifts VERY sensitive to

shell thickness.Observed shifts can be explained

with 2 nm roughness/oxide and 40% strain relaxation.

Strain level much higher than expected from critical thickness theory.

Shell Raman spectrum

Theory

ConclusionsCore -shell Raman spectra

predicted and measured.Observed strain larger than

expected from equilibrium theory.Good news! Trisilane?

Arizona welcomes you (really)

Write to us:

José Menéndez jose.menendez@asu.eduJeff Drucker jeff.drucker@asu.eduFernando Ponce fernando.ponce@asu.edu