Strain analysis of
nanostructures using
Raman spectroscopy:
theory and experiment
José Menéndez
Department of Physics
Arizona State University
Tempe, AZ 85287-USA
Just a second...
Mexico and us





30,000 Spanish refugees from Spanish War
75,000 Americans moved to Mexico fleeing
McCarthysm
50,000 Chileans fleeing dictator Pinochet
granted asylum
200,000 Salvadorans fleeing civil war
granted Mexican residency
225,000 Guatemalans fleeing counterinsurgency granted refuge
Mexicoplatz in Vienna
In 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
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
Quick Time™ a nd a
d eco mp res so r
ar e n eed ed to s ee this pi ctur e.
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
G. Liang et al., Nano Lett. 7, 642 (2007)
•Dense packing
J. Drucker
•FET performance close to balistic limit
•Wrap gates
•Core-shell: manipulation of optical and electronic
properties via index mismatch, confinement, and
strain.
Acknowledgements
Rachna Singh
Eric Dailey
Prashant Madras
Jeff Drucker
“Seedless” nanowire growth
• Grown in CVD chamber on < 1ML
Au / Si (111)
• Si wires (Si2H6) and Ge wires
(Ge2H6)
•Tunable diameter and density
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
•<011> nanowire axis direction
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
 phonon
(cm-1) 
2  kphonon
L 2 n
4 n
500 nm

0.5 nm
Why is Raman affected by
strain?


ui      K iklm  lm u k
2
0 ik
klm
Tripledegeneracy
Anharmonic coefficients.
Same symmetry as elastic
tensor
3 phonon branches here!
Strain lifts the degeneracy...

   


p xx  q  yy   zz  
2
0
2r xy

2
strained

2r xy
p yy  q  xx   zz  
2r xz
2r yz

2r xz
2r yz


0
p zz  q  xx   yy  
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
z
Phonon displacement
Sample
y
x
LightÕ
s E-field
LightÕ
s wave vector
x
u
Eout/in
uz
Eout/in
Eout/in
uy
Ein/out
Ein/out
Ein/out
Light in
Light out
Raman tensor
0 0
0 0

 0 d
0
d

0 
 0 0 d
 0 0 0


 d 0 0 
0
d

 0
uz
uy
ux
I  E  R i  Escatt
i
T
inc
0
0 0

0 0
d
2
Raman tensors
Eigenvecto
r of unperturbed
phonon j
I  E  R i  Escatt
i
T
inc
Eigenvecto
r of
perturbed
phonon i
L ji  u j  ui
Raman tensor of
perturbed phonon
i
2
Raman tensor of
unperturbed
phonon j
R i    L ji R  j 
j
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.

G. Chen et al Nano Lett. 85, 1341 (2008)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Basic equilibrium equations
 xx  yx  zx


 fx  0
x
y
z
 xy
x

 yy
y
 yz

 zy
z
 fy  0
 xz


 zz  f z  0
x
y
z
 rr  rr    1  r  zr



 fr  0
r
r
r 
z
 r 2 r 1    z



 f  0
r
r
r 
z
 rz  rz 1   z  zz



 fz  0
r
r
r 
z
Strain in cylindrical coordinates
Cartesian
1  ui u j 
 ij  


2  x j xi 
Cylindrical
u z
 zz 
z
u
 rr  r
r
1 ur ur
 

r r
r
1 ur u u
 r 


r 
r
r
1  ur uz 
1 uz u
 rz  

 z 



2
z
r
r 
z
Stress-strain relations in
isotropic materials
 
E
1  

1  1 2 
 rr 
 
 zz 











   rr   zz 
E
 1   zz    rr   

1  1 2 
z
 
1  

 rz





 r 

E
 1   rr      zz 

1  1 2 
E
1 
E
1 
E
z
rz
r
The core-shell nanowire
Si
Ge
c
a



ur r, , z  ur r


u r, , z  0



uz r, , z  uz z
Strain in core-shell nanowires
ur
ur
uz
 rr 
;   ;  zz 
r
r
z
 r  0;  zr  0;  z  0
  r  0;  zr  0;   z  0
Equilibrium equations
  rr  rr   

0

r
 r
 1  
0

 r 
  zz
 z  0

 d 2ur 1 dur ur
 0
 2 
 dr
r dr
r
 2
 d uz  0
 dz 2
Boundary conditions
No net
crosssection
force
Shrink-fit
 rrcore c   rrshell c 
 rrshell a  0

shell
zz
a
2

c 
2
c 0
core 2
zz


l  u l  l
urcore c  urshell c  c misfit
uzcore
Shrink-fit
Same
stress at
interface
shell
z
misfit
No
stress @
outer
surface
Pressing rings around shafts
Strain solution-core
E core
  shell
E

core
rr
  misfit


  11  2 




a c 

2
2
2
2
2 
1


1

2

c

1

2



a
 


a c c 




2
2


core   rrcore

core
zz




a2  c2
  misfit  2

2
2
a

c


c





Strain solution-shell
E core
  shell
E

shell
rr

shell
 misfit   1 c2

1   1  2 c2  1  2   a2

a2 
 c2  misfit
1  2   r 2   2
2
2

 a c c
 misfit   1 c2

1   1  2 c2  1  2   a2

a2 
 c2  misfit
1  2   r 2   2
2
2

 a c c

shell
zz

 c 2  misfit
r,  , z    2 2
a  c   c2



Core-shell strain
Optical phonons with strain
Phonons are the solution of the eigenvalue problem:
1,2,3 are the crystal’s cubic
cartesian coordinates
p11  q  22   33  
2r12
2r13
2r12
p 22  q 11   33  
2r 23
   2   02
2r13
2r 23
0
p 33  q 11   22  
<011> oriented nanowires
2 x 2 problem can be easily diagonalized:
 p  q  rr  q zz  
0
0
0
0
2r  zz   rr 
12 p  23 q  rr  12  p  q  zz  
2r  zz   rr 
12 p  23 q  rr  12  p  q  zz  
0
<111> oriented nanowires
3 x 3 problem can be easily diagonalized:
1
3
 p  2q 2 rr   zz  
r  zz   rr 
2
3 r  zz   rr 
2
3
r  zz   rr 
1
3  p  2q 2 rr   zz  
2
3 r  zz   rr 
2
3
r  zz   rr 
2
0
3 r  zz   rr 
1
3  p  2q 2 rr   zz  
2
3
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 interpretation
Broader, 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
44/3.7
Exp.
Theo.
+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
spectrum
Predicted 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
Conclusions
Core -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.edu
Jeff Drucker
jeff.drucker@asu.edu
Fernando Ponce fernando.ponce@asu.edu