1. Modeling of semiconductor nanowires
V.G. Dubrovskii
St. Petersburg Academic University &
Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia
Plan:
Introduction
Growth modeling
Crystal structure of III-V nanowires
Strain induced by lattice mismatch
Self-induced GaN nanowires
Self-regulated pulsed nucleation in VLS nanowires
Repino, 14 July 2013, Lecture # 2

2. Books Monograph “Theory of formation of epitaxial nanostructures”
By V.G. Dubrovskii
Moscow, Fizmatlit 2009
352 p.
New book (2013):
V.G. Dubrovskii
“Nucleation theory and growth
of nanostructures”
Springer

3. Selected papers on NWs 4 13 5

4. 8 12

5. Papers on nucleation theory
Nucleation and growth: 3
Ostwald ripening:
Linear peptide chains:

6. Modern NWs and their applications
GaN/AlN,
Ioffe & LPN CNRS
GaAs, MBE, e-beam
(Ioffe & LPN CNRS)
InAs/InP, Lund U
InAs, MOCVD,
nanoimprint (Lund U)
Nanosensors
Nanophotonics
Nanoelectronics

7. Modern nanowires and their importance
Nano Lett. 10, 1529 (2010)
Where is the
killer application?
Nanowire based single cell endoscopy
Exponential increase in the number of
publications:
Biological probe for endoscopy, spot delivery and
sensing within a single living cell
2) Nanowires for direct solar to fuel conversion
3) Integrated nanophotonics
1 – solar cells, 2 – LEDs/lasers, 3 – nanoribbons,
4 – photonic bandgap NW arrays, 5 – sample analysis chambers, 7- photodetectors, - microfluidic systems
4) fundamental physics: growth and properties

8. Advantages of nanowire based optoelectronics
Easy to fabricate uniform arrays by organizing seeds before growth
Smallest LEDs / lasers of any kind (10s nm in diameter, a few microns in length)
(Potentially) high efficiency (electronic active medium and
optical waveguide being identical: large confinement factor)
Vertical cavity and surface emitting
Easy to realize single photon emission
Much less restricted by lattice mismatch => III-Vs on Si substrates, coherent strained heterostructures in NWs
Wurtzite phase of ZB III-Vs
(C. Chang-Hasnain group, UC Berkeley APL 2007)

9. Nanowire heterostructures

10. Au-assisted VLS growth: the first wires
Au-assisted CVD of Si “whiskers” on Si(111) at T~1000 0C (Wagner & Ellis, 1964)
Fundamental aspects of VLS growth: Givargizov, in “Highly anisotropic crystals”, 1975
Alloy at
equilibrium
with solid
Liquid
Vapor
Au catalyst
Si is transferred from vapor to solid through liquid drop on the
wire top (Tm=363 0C)
Liquid drop acts as a chemical catalyst: pyrolysis rate > 0 at
the drop surface and = 0 at the substrate surface
Simple phase diagram of Au-Si alloy: no Au in the wire?
Si wires

11. Au-assisted VLS growth of III-V nanowires by MBE

12. Kinetic processes driving nanowire growth
1 – direct impingement
2 – desorption from the drop
3 – diffusion from the sidewalls
4 – desorption from the sidewalls
5 – diffusion from the substrate
to the sidewalls,
6 – diffusion from the substrate to
the drop
7 – surface nucleation
Nucleation-mediated wire growth
resulting in the vertical
growth rate
Supersaturation of gaseous phase to the solid
(= to equilibrium alloy with concentration Ceq)
Supersaturation of (liquid) alloy in the drop to the solid
V.G.Dubrovskii et al., PRE 2004, PRB 2005, PRE 2006, PRB 2008, PRB 2009; PRB 2010, APL 2011
W.Seifert et al., JCG 272, 211 (2004), L.Schubert et al., APL 84, 4968 (2004)….

13. Model of diffusion-induced NW growth
Stationary growth with R = const
Direct impingement
Adatom diffusion, substrate and sidewalls
GT effect in the drop
Surface adatoms (s):
Sidewall adatoms (f):
Four boundary conditions:
ω = 1 in MOCVD and 1/π in MBE
Constant concentration
far away from the wire
Continuity of chemical
potential at the wire base
Continuity of flux
at the wire base
Continuity of chemical
potential at the wire top
V.G.Dubrovskii et al., PRB 2005, 2009, PRE 2006

14. Growth kinetics Due to GT effect, coefficients
A, B and C can be of either signs !
Direct impingement,
Surface growth
Sidewall
adatoms
Surface α β J Sa
B=0, C=0 (no diffusion): dl/dh=A, Classical Givargizov-Chernov case
Generally:
DI growth:

15. Theoretical L(t) curves
@ RGT=3.5 nm
Au-assisted MBE
of GaAs NWs
L(t) curves are essentially
non-linear !!!
V.G. Dubrovskii et al., PRB 2009

16. Narrowing size distribution of <110> Ge NWsnm
Initial stage:
Infinite growth:
Limited growth:
Dubrovskii et al., PRL 2012

17. Role of surface energies in NW polytypism
Hexagonal
cross-section:
1-st approximation
for lateral
surface energy:

18. Surface energies: summary
Surface energy ratio WZ to ZB
Facet type
Surface energy,
J/m2
Transition
1.73
0.75
1.50
0.867
1.30 1
Dubrovskii et al., PRB 2008; Phys. Solid State 2010

19. Role of nucleation
F.Glas et al., Phys. Rev. Lett. 2007, V.G. Dubrovskii et al., PRB 2008, J. Johansson et al,
Cryst. Growth & design 2009 …
At lower surface energy of NW sidewalls, WZ phase can form only when nucleation
takes place at the triple phase line (TL)
In a mononuclear mode, the structure is dictated by the monolayer island orientation
Two conditions of WZ phase formation:
Condition for TL nucleation (straight sidewalls):
LV surface energy should not be too high!
High enough supersaturation to create a
stacking fault
G2< G1
Nucleation barriers

20. Theoretical conclusions
Surface energy of relevant WZ sidewalls is indeed lower than of ZB ones
TL nucleation can be suppressed by a lower surface energy catalyst
Structure retains to bulk ZB at large R because the ring of critical size
dissapears (Dubrovskii et al., PRB 2008)
Growth and phase diagrams:
CUB – blue curves
HEX – red curves
0.83
0.875
τ=0.95
0.91
GaAs, R=20 nm

21. Two step growth with temperature ramping
1 nm Au layer deposited on GaAs(111)B surface
Sample A grown at 6300C from the beginning => no NW growth
Growth at 5300C for tLT; growth temperature ramped
from 530 to 6300C within 2 min, Ga and As4 fluxes maintained;
growth at 6300C, V=0.2 nm/s
Sample B: tLT=1.5 min, NO NW growth
Sample C: tLT=15 min, NW GROW longer than 2 nm
V.G. Dubrovskii et al.,
PRB 2009
Riber 32 (LPN)
Complex NW shape:
Branching, tapering
Continuing growth
Sample A: tLT=0
Sample B: tLT=1.5 min
Sample C: tLT=15 min

22. Two step growth with temperature ramping
- 0.4 nm Au layer deposited on GaAs(111)B surface
- Samples 1 grown at 6300 C from the beginning => no NW growth
Growth at 5500 C with V=0.3 nm/s for tLT
- Growth temperature ramped from 530 to 630 0C,
- Growth at 6300 C for 48 min, V=0.15 nm/s, V/III=4.
Sample 2: tLT =2 min, NO NW growth -
Sample 3: tLT =12 min, NW GROW longer than 10 microns
EP1203 (Ioffe)
More regular shape
Sample 3 before and after high temperature growth step

23. Control of crystal structure: stacking fault free GaAs NWs grown with two T steps via scenario IV
High resolution TEM studies of a NW detached from sample C:
200 nm
40 nm
Pure WZ
Pure ZB
Transition region

24. Optical properties of WZ and WZ/ZB GaAs NWs:
Pure WZ NWs
B.V. Novikov et al., PSS RRL 2010:
WZ/ZB heterostructures
D. Spikoska et al., PRB 2009:
Type II band structure:
Predominantly ZB NWs
Pure WZ NWs
Band alignment and the first
e and h levels v thickness
PL spectra: 1.51 to
1.43 eV shift for different
proportions of WZ
EZB-EWZ=41 meV, redshift opposite to InP

25. Control of crystal phase by growth catalyst: Ga-catalyzed GaAs NWs
Dubrovskii et al., PRB 2010, Nano Letters 2011
TPL nucleation condition:
1.3 J/m2
J/m2
J/m2
for Au-Ga
(at 40% Ga percentage)
= to J/m2
for contact angles from 110 to 1250
A=drop
B=WZ
C=WZ-ZB
mix-up
D=ZB all the way
For pure liquid Ga at the
growth temperature: =
0.08 to 0.16 J/m2

26. Strain relaxation and critical dimensions in freestanding nanowires
Because of free lateral surfaces
strain relaxation is expected to be
much more efficient
than in 2D layers and even QDs
Model
- linear isotropic elasticity
- same elastic parameters E, n
Barton J. Appl. Mech. (1941) 
Elastic modulus
Poisson ratio

27. Strain maps
ezz /e0
axis
outer surface
E = 90 GPa, ε0 = 0.46, ν = 0.3

28. Heterostructured nanowires (QDs in NW)
Because of free lateral surfaces
strain relaxation much more efficient
than in 2D layers and even QDs !!!
InAs QDs in InP NWs:
Axial or radial heterostructures
Lund University

29. Relaxation of elastic stress in NS grown on a lattice mismatched substrate: existing models
Major asymptotic properties:
Simple:
Elastic energy of 2D layer
(per atom)
Ratsch-Zangwill:
Aspect ratio:
Glas:
Gill-Cocks:

30. Results for elastic energy relaxation
Elastic constants of a cubic material
Solid lines – calculations for different geometries
Dashed lines - fits
Relative strain energy for cylinders
5.5 (cone), 8 (truncated cone 700), 15
(cylinder) and 50 (reverse cone 1100)

31. Critical thickness for plastic relaxation
Energy per of a dislocation pair (Glas, PRB 2007): z r h 2R if if
b is the core cutoff parameter for elastic stress, θ is the angle
between the Burgers vector and dislocation line
Elastic energy:
a = 1 for cylinder and 0 for cone

32. Critical thickness for plastic relaxation
The excess energy of dislocation pair with respect to a fully coherent state is:
Pure edge dislocations:
600 dislocations:
Coherent state is stable
Dislocations
Critical thickness for dislocation formation

33. Critical thickness for plastic relaxation
600 dislocations in cylinder geometry:
Critical thickness tends
to infinity at certain
critical radius which
depends on lattice
mismatch and NS
geometry!
4% - GaAs/Si, 8.1% - InP/Si, 11.6% - InAs/Si

34. Critical dimension for plastic deformation
, therefore the equation for critical dimension is given by
Critical radius v mismatch for different
geometries:
Dots showing MOCVD
and MBE experimental data

35. III-V NWs on Si substrates: MOCVD
a – InAs with 20 nm Au on Si(111)
b – InP with 20 nm Au
c – InP with 60 nm Au
d – InP with 120 nm Au
e – TEM of 17 nm diameter InAs NW
Critical diameter for the growth of epitaxial NWs
on the lattice mismatched substrates
(C. Chang-Hasnain group, APL 2007)
WZ phase !!!

36. III-V NWs on Si substrates: MBE
Cirlin et al., PSS RRL 2010

37. Problems with VLS nanowires
Unwanted Au contamination
Uncontrolled zincblende-wurtzite polytypism
Use catalyst-free NN formation (GaAs on Si or sapphire)
Use self-catalyzed growth (Ga instead of Au
in the case of GaAs NWs)
Au distribution in Au-seeded Si NWs
(by P. Pareige, Rouen University,
France)

38. Au contamination of Si and Ge NWs grown by MBE
Nanoscale RL

39. Self-induced GaN NWs on Si: new growth mechanism
No Ga drops are detected on top => not VLS mechanism
GaN never nucleates as NW, nanoislands of different shapes are
formed in the beginning (different shapes on an amorphous SixN
interlayer or on mismatching AlN layer)
Even on AlN, misfit dislocations are formed before NW formation; NWs
are relaxed from the very beginning
MBE of self-induced GaN NWs employs specific growth conditions:
high N flux and high temperature are required
Surface diffusion plays a crucial role in NW growth
GaN NWs usually grow in both vertical and radial directions
GaN NWs are hexahedral, restricted by 6 equivalent low energy m-planes

40. Self-induced GaN NWs on Si(111): radial growth !
Histograms showing
diameter distributions:
Growth mechanism:
Length-diameter dependence:
No drops are seen on NW tops
NWs growing in vertical and radial direction

41. Nucleation on lattice mismatched AlN layer
MBE on Si(111) substrates
5 nm thick AlN buffer layer
GaN growth at T=800 C, N/Ga fluxes ratio =10
HR TEM images:
RHEED patterns:
2 min, AlN buffer
10 min, GaN islands
17 min, GaN NWs
RHEED and HRTEM studies show misfit
dislocations in islands!
a – SC islands; b – truncated pyramids
c – full pyramids, d – NWs,
island to NW transition at ~ nm radius

42. Role of misfit dislocations
Height v radius for different structures:
dislocation

43. GaN NWs are relaxed from the beginning!
Model suggesting a series of shape transformations
to relax elastic stress, NW is already relaxed:
Plastic relaxation in islands is also shown in:
O. Landre, C. Bougerol, H. Renevier, and
B. Daudin. Nanotechnology 20, (2009)

44. Nucleation on an amorphous interlayer
Si(111) substrate
5 min exposure to active N to form SixNy amorphous layer
GaN growth at T=780 C, N/Ga fluxes ratio =6.2
Epitaxial constraint should be weak!
HR TEM:
RHEED patterns:
r0=5 nm
Incubation
Transition SC
NWs

45. Scaling model for nucleation and growth of GaN NWs
J. Tersoff, R.M. Tromp, Phys. Rev. Lett. 70, 2782 (1993)
Assumptions:
No strain-induced contributions, directly applicable on an amorphous interlayer
Anisotropy of surface (and edge) energy as the dominant driving force
Growth anisotropy: superlinear length-radius dependence of GaN NWs !
Compare surface energy of isotropic island and anisotropic NW at given volume
Illustration of the model:
Surface energy of isotropic island:
In SC geometry:
Island volume:
In SC geometry:
NW volume:
Surface energy of NW:

46. Scaling model for h(r)
Superlinear dependence of NW length on radius with a>1 for all t
With this dependence,
from
Using this in previous equations, the
driving force for island to NW shape
transformation is obtained in the form
sidewalls
edges
in-plane
Results of statistical analysis of TEM and
SEM data remarkably follows the scaling
dependence at:
and

47. General condition for anisotropic growth
NW anisotropic growth is energetically preferred
NW growth is suppressed
No edge contributions
between
where
are positive roots of cubic equation
Interesting NW case relates to
NW sidewall energy should be much
smaller and in-plane energy compared
to surface energy of the island !
0.088

48. Parameters of GaN spherical caps and NWs
Boxy hexahedral islands with
constant aspect ratio h/r = 0.088
5 nm from experimental data
3.4 nm from growth law
In view of small prefactor
and larger contact angle of NWs
130 meV/A2
known
137 meV/A2
Assume
100 meV/A2
meV/A2 (was 40 meV/A2 by analogy with Si/SiO2)
230 meV/A2 (was meV/A2 from Young’s eq.)
0.088

49. Scaling in GaN NW growth: kinetic model
Elongation:
Tip SW surface
Top facet
Radial growth:
Schematics of possible growth scenarios:
Yellow – NW surface contributing to elongation
Magenta – desorption area
Grey – NW surface contributing to radial growth
Blue – overgrown shells
SW collection
a – no radial growth, R=const
b – R~t
c – tapered shape
d – cylindrical shape, SCALING!
Neglect c, adopt model d with

50. Scaling in GaN NW growth: L(t), R(t)
V=0.045 nm/s; a=65 nm
R(t0)=17 nm, L(t0)= 140 nm
d=2.46:
Condition for super-linear NW growth:

51. Timescale hierarchy and self-regulated pulsed nucleation in catalyzed nanowire growth
V.G. Dubrovskii
St. Petersburg Academic University &
Ioffe Institute RAS, St. Petersburg, Russia
Plan:
Nucleation statistics
Oscillating morphology of growth interface
Sharp nucleation probability: impact on length uniformity
Nucleation theory applied to monolayer growth cycle
Timescale hierarchy
Au-catalyzed GaAs nanowires
Conclusions
V. G. Dubrovskii Phys. Rev. B. 87, (2013)
Lecture 3, Repino , 14 July

52. Usual assumptions Droplet is liquid
Supersaturation in the droplet is constant during growth.
Liquid-solid growth interface is planar
From Dubrovskii & Sibirev JCG 2007:
From Glas et al. PRL 2007:

53. Nucleation statistics in InPAs nanowires
Post-growth study of compositional modulated InPxAs1-x wires:
Au-catalyzed MBE
– HAADF STEM image showing composition
oscillations, related to a given time interval
(b) – measured L(t) fitted by the diffusion
growth model
Experimental determination of nucleation statistics
– Length of successive nucleations, dashed line
is the mean height and solid line is the mean length
(b) – Histogram of nucleation events per osilattion
Blue line – Poissonian; Red line – model of Glas
Std deviation
v length:

54. Periodically changing morphology of the growth interface in catalyzed Si, Ge and GaP nanowires
If a truncated facet is stable:
Sawtooth

55. Cyclic supersaturation in Au-catalyzed Ge nanowire growth
2011

56. Impact on the length distribution of nanowires
Regular NW arrays with L=const: if droplets are organized before growth,
then the wires have a narrow distribution over L
L=const for R=const!
Probability to observe a
NW with m MLs at time t
Au-seeded InAs,
MOCVD, nanoimprint
Au-seeded GaAs,
MBE, e-beam
Hypothetical growth from identical droplets,
starting simultaneously at t=0, with average
growth rate V (in ML/s), and RANDOM nucleation:
Poissonian length distribution

57. Evolution of length distribution with growth time
Why this unwanted Poissonian broadening is not observed experimentally?

58. Material balance within 1 ML growth cycle
Consider an element that limits nucleation (Si in Au-Si or As in Au-Ga-As).
Supersaturation:
Perfect alloy approx.
Atomic concentration of As in the droplet
Droplet volume
Linear scaling:
Model system:
Formation of 1 ML removes As
atoms from the droplet
Refill time:
Deposition rate
Effective diffusion length on NW sidewalls

59. Nucleation and growth of 2D island
The probability density of island nucleation:
Re-normalized Zeldovich nucleation rate
Island growth rate:
Maximum
suoersaturation
Material balance (in absence of desorption):
Atoms dissolved
in the droplet
Number of atoms
In 2D island
Total number of atoms arrived
to the droplet by time t
Analytical solutions:
!!!
Probability density
Nucleation probability

60. Time scale hierarchy Island growth time = Maximum supersaturation
Analysis for Au-catalyzed GaAs:
Nucleation to refill:
Growth to nucleation:
0.045 nm3
0.326 nm
450 to 600 C,
0.35 J/m2
0.005
Island growth << Nucleation interval << Refill
R=25 nm

61. Radius dependence Same parameters of GaAs NWs
Non-overlapping probabilities: narrow nucleation pulses, anti-correlation, uniform L
Overlapping probabilities: random Poissonian
Pulsed nucleation requires (i) modest growth rate; (ii) fast diffusion in the liquid;
(iii) small enough radius

62. Impact of truncated edge on crystal structure
When main facet does not meet the trijunction, islands do not nucleate at the trijunction!
Wetting VLS growth predicted in
Experimental verification of wetting:
C. García Núñez et al. J. Cryst. Growth 372, 205 (2013)
Wetting without or with a truncated
facet has a very similar impact on
the crystal structure:

63. Thank you very much