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
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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.
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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 [email protected] Drucker [email protected] Ponce [email protected]
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