Modeling of semiconductor nanowires

V.G. Dubrovskii

St. Petersburg Academic University & Ioffe Physical Technical Institute RAS, St.-Petersburg, Russia

Repino, 14 July 2013, Lecture # 2

dubrovskii@mail.ioffe.ru

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

Books

Monograph“Theory of formation of epitaxial nanostructures”By V.G. DubrovskiiMoscow, Fizmatlit 2009352 p.

New book (2013):V.G. Dubrovskii“Nucleation theory and growthof nanostructures”Springer

Selected papers on NWs

5

413

8

12

Papers on nucleation theoryNucleation and growth:

Ostwald ripening:

Linear peptide chains:

3

Modern NWs and their applications

InAs, MOCVD, nanoimprint (Lund U)

GaAs, MBE, e-beam(Ioffe & LPN CNRS)

InAs/InP, Lund UGaN/AlN,

Ioffe & LPN CNRS

NanoelectronicsNanophotonics

Nanosensors

NW

Modern nanowires and their importanceNano Lett. 10, 1529 (2010)

Exponential increase in the number of publications:

Where is the killer application?

1) Nanowire based single cell endoscopy

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

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)

Nanowire heterostructures

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

Au catalyst

Si wires

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?

Alloy at equilibriumwith solid

LiquidVapor

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

Kinetic processes driving nanowire growth

Substrate

Surface layer

Island

2R

1

Wire

2

3 6

4

5 7

LL0

Hs

VL

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 tothe drop7 – surface nucleation

Nucleation-mediated wire growth resulting in the vertical growth rate

1/1/ surfdes

eqdes VVVV

1/ eqCC

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)….

Model of diffusion-induced NW growth

λs

λf

α

L

2RJ

l

θs

θf

r

z

β

γ

0cos s

ssss

nJnD

• Stationary growth with R = const• Direct impingement• Adatom diffusion, substrate and sidewalls• GT effect in the drop

Surface adatoms (s):

Sidewall adatoms (f):

0sin2

2

f

ff

ff

nJ

dz

ndD

ω = 1 in MOCVD and 1/π in MBEFour boundary conditions:

0r

s

dr

dn Constant concentration far away from the wire

0

z

ff

Rr

ss dz

dnD

dr

dnD Continuity of flux

at the wire base

)0()( ffss nRn Continuity of chemical potential at the wire base

R

LnTk llffB

2

)(ln

Continuity of chemical potential at the wire topV.G.Dubrovskii et al., PRB 2005, 2009, PRE 2006

Growth kinetics

fLl / fHh /

AlU

ClBU

dh

dl

)(

)(

0)0( lhl

Direct impingement,Surface growth

Sidewalladatoms

Surfaceadatoms

B=0, C=0 (no diffusion): dl/dh=A, Classical Givargizov-Chernov case

ACdh

dl

l

0

Generally:

ABdh

dl

l

DI growth: 1/ Rs 1tan)/( Rf

CBUdh

dU

ClBU

ClBU

Blh

)(

)(ln

1)(

0

Due to GT effect, coefficientsA, B and C can be of either signs !

)()0( 0lUhU

α

β

J

Sa

1)cosh()sinh()( lllU

Theoretical L(t) curves

0 1000 2000 3000 4000 5000 6000

200

400

600

800

1000

1200 R=50 nm

R=30 nm

R=20 nm

L [n

m]

t [s]

R=10 nm

Au-assisted MBE of GaAs NWs

L(t) curves are essentiallynon-linear !!!

V.G. Dubrovskii et al., PRB 2009

@ RGT=3.5 nm

Narrowing size distribution of <110> Ge NWs

60 80 100 120 140 160 1800

500

1000

1500

2000

2500

40 min

70 min

50 min

30 min25 min

Leng

th (

nm)

Diameter (nm)

15 min

Initial stage: RttVL s /)(2 0

Infinite growth: ls 0lf 0

Limited growth: ,0sg 0fg

100~s nm

1

2exp

2

R

Hag

abg

gL f

f

ss

Dubrovskii et al., PRL 2012

Role of surface energies in NW polytypism

ii

ii

wv S

n

22)112( 23.10

7819.0

8

nmnm

22)0011( 67.7

7822.0

6

nmnm

22)011( 86.8

4514.0

4

nmnm

2

2)0211( 86.84516.0

4

nmnm

Hexagonal cross-section:

1-st approximationfor lateral

surface energy:

Surface energies: summary

Facet typeSurface energy,

J/m2 Transition

1.73 0.75

1.50 0.867

1.30 0.867

1.50 1

)112(

)011(

)0011(

)0211(

)112( )0011(

ZBWZ /

)112( )0211(

)011( )0011(

)011( )0211(

Surface energy ratio WZ to ZB

Dubrovskii et al., PRB 2008; Phys. Solid State 2010

Role of nucleation

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

F.Glas et al., Phys. Rev. Lett. 2007, V.G. Dubrovskii et al., PRB 2008, J. Johansson et al, Cryst. Growth & design 2009 …

Two conditions of WZ phase formation:

r*1r*2

G*1

0

G

G*2

r

12 2<

Condition for TL nucleation (straight sidewalls):

lSLLVWV sin

WV

C nucleation

LV

lSL

(a)

TL nucleation

(b)

SL

High enough supersaturation to create astacking fault

LV surface energy should not be too high!

2* jjG Nucleation barriers

2)/(1 ZBWZ

WZc

Theoretical conclusions

TL nucleation:ρ*

max = ρC nucleation:ρ*

max = 3ρ/2

3ρ/2

3

4

5

6

7

8

CUBfmin

HEX,TL

fmin

CUB,TL

fCR

0 1.51.2510.750.5

No

rma

lize

d c

he

mic

al p

ote

ntia

l f

fmax

0.25

Normalized wire radius

HEX

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)

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.00.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

litie

s p C

UB,

p HE

X

Liquid chemical potential f

Growth and phase diagrams:CUB – blue curvesHEX – red curves0.83 0.875

0.91

τ=0.95

GaAs, R=20 nm

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

Sample A: tLT=0 Sample B: tLT=1.5 min Sample C: tLT=15 min

Continuing growth

Riber 32 (LPN)

Complex NW shape:Branching, tapering

V.G. Dubrovskii et al.,PRB 2009

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

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:

Pure WZ Pure ZB

40 nm

Transition region

200 nm

Optical properties of WZ and WZ/ZB GaAs NWs:WZ/ZB heterostructuresD. Spikoska et al., PRB 2009:Type II band structure:

Band alignment and the first e and h levels v thickness PL spectra: 1.51 to

1.43 eV shift for differentproportions of WZ

Pure WZ NWsB.V. Novikov et al., PSS RRL 2010:

Predominantly ZB NWs

Pure WZ NWs

EZB-EWZ=41 meV, redshift opposite to InP

Control of crystal phase by growth catalyst: Ga-catalyzed GaAs NWs

A=dropB=WZC=WZ-ZB mix-upD=ZB all the way

0sin LVSLWV

TPL nucleation condition:

WV 1.3 J/m2

59.0SL J/m2

0.1LV J/m2 for Au-Ga (at 40% Ga percentage)

= -0.23 to -0.11 J/m2

for contact angles from 110 to 1250

For pure liquid Ga at the growth temperature:

67.0LV

= 0.08 to 0.16 J/m2

Dubrovskii et al., PRB 2010, Nano Letters 2011

Strain relaxation and critical dimensionsin freestanding nanowires

01 a

a

a

• 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,

Barton J. Appl. Mech. (1941)

Elastic modulus

Poisson ratio

Strain maps

zz /0

axis outer surface

E = 90 GPa, ε0 = 0.46, ν = 0.3

Heterostructured nanowires (QDs in NW)

InAs QDs in InP NWs:

Lund University

Because of free lateral surfaces strain relaxation much more efficient than in 2D layers and even QDs !!!

Axial or radial heterostructures

Relaxation of elastic stress in NS grown on a lattice mismatched substrate: existing models

0/1

1)(

z

)()( 2 zww D 1)( 0 z /)( 0 z

)1/()( 202 Ew D Elastic energy of 2D layer

(per atom)

Major asymptotic properties:

Simple:

0

0

/

)/exp(1)(

z

Ratsch-Zangwill:

)exp()1(1

)( 312

1

ppp

pz

Glas:

22

2

)1016()61(21

61)(

kbz

Gill-Cocks:

)2/( RHAspect ratio:

Results for elastic energy relaxation

DWWz 2/)()(

Solid lines – calculations for different geometries

Dashed lines - fits )1/(1)( AZ

A 5.5 (cone), 8 (truncated cone 700), 15 (cylinder) and 50 (reverse cone 1100)

Elastic constants of a cubic material

Relative strain energy for cylinders

Critical thickness for plastic relaxation

Energy per of a dislocation pair (Glas, PRB 2007):

1ln)1(8

)cos1(4

_

2

22

b

h

v

bvERWd

zr

h

2R

hh _

Rh Rh _

Rh /2if if

b is the core cutoff parameter for elastic stress, θ is the angle between the Burgers vector and dislocation line

Elastic energy:

3

1

11)(

1

220

220

aa

A

hR

v

EVZ

v

EWe

a = 1 for cylinder and 0 for cone

Critical thickness for plastic relaxation

The excess energy of dislocation pair with respect to a fully coherent state is:

1

),(ln

3

1

421),(

_2

02

22

0

b

hRhC

aa

R

b

R

bRh

R

hZ

v

ErhRW effeff

)]1(2/[)cos1( 22 vbvC /4

Pure edge dislocations: 2/ bbeff

600 dislocations: 3/ 2/bbeff

0W Coherent state is stable 0W Dislocations

0),( hRW Critical thickness for dislocation formation )(Rhc

Critical thickness for plastic relaxation

4% - GaAs/Si, 8.1% - InP/Si, 11.6% - InAs/Si

03.00 600 dislocations in cylinder geometry:

Critical thickness tendsto infinity at certain critical radius which depends on lattice mismatch and NS geometry!

Critical dimension for plastic deformation

AZ /1)( at , therefore the equation for critical dimension is given by

01ln43

)1(20

222

b

RCRb

b

A

aa cceff

eff

Critical radius v mismatch for differentgeometries:

Dots showing MOCVDand MBE experimental data

III-V NWs on Si substrates: MOCVD

Critical diameter for the growth of epitaxial NWson the lattice mismatched substrates(C. Chang-Hasnain group, APL 2007)

a – InAs with 20 nm Au on Si(111)b – InP with 20 nm Au c – InP with 60 nm Aud – InP with 120 nm Au e – TEM of 17 nm diameter InAs NW

WZ phase !!!

III-V NWs on Si substrates: MBE

Cirlin et al., PSS RRL 2010

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)

Au contamination of Si and Ge NWs grown by MBE

Nanoscale RL

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

Self-induced GaN NWs on Si(111): radial growth !

Growth mechanism:

Length-diameter dependence:

Histograms showing diameter distributions:

No drops are seen on NW tops NWs growing in vertical and radial direction

Nucleation on lattice mismatched AlN layer

RHEED patterns:

2 min, AlN buffer

10 min, GaN islands

• MBE on Si(111) substrates• 5 nm thick AlN buffer layer• GaN growth at T=800 C, N/Ga fluxes ratio =10

17 min, GaN NWs

HR TEM images:

a – SC islands; b – truncated pyramidsc – full pyramids, d – NWs, island to NW transition at ~ 13-14 nm radius

RHEED and HRTEM studies show misfitdislocations in islands!

Role of misfit dislocationsHeight v radius for different structures:

dislocation

GaN NWs are relaxed from the beginning! Model suggesting a series of shape transformationsto 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, 415602 (2009)

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!

RHEED patterns:

Incubation SC TransitionNWs

HR TEM:

r0=5 nm

Scaling model for nucleation and growth of GaN NWs

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:

02

0)( rkrkkG ISLn

siinnISL

In SC geometry:

n

SCnnk )cos1/(2 ik 2k

Island volume: 30rkV VISL

In SC geometry: 3/)]([ fkV

]sin)cos1/[()]cos2)(cos1[()( f

Surface energy of NW:

J. Tersoff, R.M. Tromp, Phys. Rev. Lett. 70, 2782 (1993)

rrrhG NWSiTOPSWNW 6)(2

336 2

NW volume:

hrVNW2)2/33(

Scaling model for h(r)

rh Superlinear dependence of NW length on radius with >1 for all t

Results of statistical analysis of TEM and SEM data remarkably follows the scalingdependence at:

46.2 088.0and

With this dependence,from NWISL VV

3

23/1

0 2

33

r

kr

V

Using this in previous equations, the driving force for island to NW shape transformation is obtained in the form

3/)2(2/)()(

rAGGrg ISLNW

1)( 3

)2(

3

)12(

3

)1(2

3

1

rerdrcrbrg

ABb / ACc / ADd / AEe /

nsiinn

V

kkk

A )(2

333/2

sidewalls

SWB 6

)(2

33SiTOPC

in-plane edges

NWD 6 ISLV

kk

E

3/1

2

33

General condition for anisotropic growth

0)( rg NW anisotropic growth is energetically preferred

0)( rg NW growth is suppressed

0 edNo edge contributions 0)( rg between )1/(32,12,1

xr

where 2,1x are positive roots of cubic equation 023 cxxb

27/42 cb

Interesting NW case relates to

1~1r 12 r

1b 1~c

NW sidewall energy should be much smaller and in-plane energy compared to surface energy of the island ! 0 10 20 30 40 50 60 70

-0,50

-0,25

0,00

0,25

0,50

b=0.34

b=0.12

g (r)

r (nm)

c=0.7

14d 5.4e

46.2 0.088 7.0c

Parameters of GaN spherical caps and NWs

0 10 20 30 40 50 60 70-0,75

-0,50

-0,25

0,00

0,25

0,50

r2

Edge terms included

g1(r)

g(r)

NW radius r (nm)

g(r)

No edge terms

r1

Boxy hexahedral islands withconstant aspect ratio h/r = 0.088

46.2 0.088

14.0b 7.0c14d 5.4e

0r 5 nm from experimental data

1r 3.4 nm from growth law

14.0b 7.0c

14d 5.4e

ed In view of small prefactor

and larger contact angle of NWs

TOP 130 meV/A2

S 137 meV/A2 known

Assume SW 100 meV/A2

109i meV/A2 (was 40 meV/A2 by analogy with Si/SiO2)

SC 230 meV/A2 (was 130-176 meV/A2 from Young’s eq.)

Scaling in GaN NW growth: kinetic model

Schematics of possible growth scenarios:Yellow – NW surface contributing to elongationMagenta – desorption areaGrey – NW surface contributing to radial growthBlue – overgrown shells

nmLL 40~~0

22

)cos(2sin

RJJJRJJ

dt

dLRsurfdestoptop

f

RLJ

J

dt

dRRLSW

f 2sin2

Elongation:

Tip SW surface Top facet

Radial growth:

SW collection

a – no radial growth, R=constb – R~tc – tapered shaped – cylindrical shape, SCALING!

cR

a

dt

dL

V

1B

dt

dR

V

1

Neglect c, adopt model d with Lconst /00)( RttR 00)( LttL

Scaling in GaN NW growth: L(t), R(t)

R

a

dt

dL

V

1

L

b

dt

dR

V

1

/)tan2( SWf gb

/)tan2( ff ga )sin/()(1 JJg ftopf

)sin/()(1 JJg fSWSW

)1/(

00

00

)()1(1

RL

ttVaLL

)1/(1

00

00

)()1(1

RL

ttVaLR

SWf

topf

SW

f

JJ

JJ

g

g

b

a

sin

sin

00 R

RLL

Condition for super-linear NW growth: topstep JJ

V=0.045 nm/s; a=65 nmR(t0)=17 nm, L(t0)= 140 nm=2.46:

0 5000 10000 15000 20000 250000

500

1000

1500

2000

0

10

20

30

40

50

60

NW

leng

th (

nm)

Growth duration (s)

NW

rad

ius

(nm

)

Timescale hierarchy and self-regulated pulsed nucleation in catalyzed nanowire growth

V.G. DubrovskiiSt. Petersburg Academic University &

Ioffe Institute RAS, St. Petersburg, Russia

Lecture 3, Repino , 14 July

dubrovskii@mail.ioffe.ru

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, 195426 (2013) 

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:

Nucleation statistics in InPAs nanowiresPost-growth study of compositional modulated InPxAs1-x wires:

Std deviationv length:

Au-catalyzed MBE

(a) – 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(a)– Length of successive nucleations, dashed line is the mean height and solid line is the mean length(b) – Histogram of nucleation events per osilattionBlue line – Poissonian; Red line – model of Glas

Periodically changing morphology of the growth interface in catalyzed Si, Ge and GaP nanowires

10y

y

)/(

1

10 MLttaay

Sawtooth )(t

If a truncated facet is stable:

Cyclic supersaturation in Au-catalyzed Ge nanowire growth

2011

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

Au-seeded InAs, MOCVD, nanoimprint

Au-seeded GaAs, MBE, e-beam

L=const for R=const!

Hypothetical growth from identical droplets, starting simultaneously at t=0, with average growth rate V (in ML/s), and RANDOM nucleation:

00 Vp

dt

dp

)(tpmProbability to observe a NW with m MLs at time t

)( 1 mmm ppV

dt

dp ...3,2,1m

Poissonian length distribution

!

)()(

m

Vtetp

mVt

m

Evolution of length distribution with growth time

0 20 40 60 80 100 1200.000

0.025

0.050

0.075

0.100

0.125

100Vt

50Vt

Pro

babi

lity

p m(t

)

Number of monolayers in nanowire

10Vt

Why this unwanted Poissonian broadening is not observed experimentally?

Material balance within 1 ML growth cycle

Consider an element that limits nucleation (Si in Au-Si or As in Au-Ga-As).

Supersaturation:

Atomic concentration of As in the droplet

constRfN 30 )( Droplet volume

Formation of 1 ML removes

hRN /2 As atoms from the droplet

)(

Rfch

eq

]/)cos1[(2

11 RV

tr Refill time:

hIV / Deposition rate

)/(Vhjdiff Effective diffusion length on NW sidewalls

Model system:

)1ln( TkB

1/ 0 NNc

)/( 0 eqcNNLinear scaling:

1/ eqcc Perfect alloy approx.

Nucleation and growth of 2D island

)()/( 2 tPRh The probability density of island nucleation:

Re-normalized Zeldovich nucleation rate

1)/( 2 Tkha B Island growth rate: // 0rdtdr

Material balance (in absence of desorption):

2

0

22 )()()(2

cos12

)0(

t

t

t

diff

ttdtPtd

RhtNtRj

IRN

*)0(

Total number of atoms arrived to the droplet by time t

Atoms dissolvedin the droplet Number of atoms

In 2D island

Maximum suoersaturation

0)0( tP

nn t

tt

t

ttPt **

* expexp)()(

Analytical solutions:

1)/ln(3

exp*

t

a

nt

tttQ *expexp1)(

/)( *ttn

1)(~ * ci !!!

)]1ln(/exp[ aP

Probability density Nucleation probability

Time scale hierarchy

gt Island growth time = )/( 0*rR

)cos1/(2

)( 2

R

R

hV

fct eqMaximum supersaturation *

h

Rfc

t

t eq

r

n)(

*

02* r

R

tt

t

n

g

Nucleation to refill: Growth to nucleation:Analysis for Au-catalyzed GaAs:

rng ttt 2 Island growth << Nucleation interval << Refill

0.045 nm3 h 0.326 nm

450 to 600 C, 0.35 J/m2

40a eqc 0.005 2/ 0

sMLV /5

R=25 nm

Radius dependence

Same parameters of GaAs NWs

0.00 0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

Pro

babi

lity

dens

ity P

(t)

Time t (s)

R=25 nm R=150 nm

0.00 0.05 0.10 0.15 0.200.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

ty Q

(t)

Time t (s)

R=25 nm R=150 nm

4102 V

5105 V

Non-overlapping probabilities: narrow nucleation pulses, anti-correlation, uniform LOverlapping probabilities: random Poissonian

Pulsed nucleation requires (i) modest growth rate; (ii) fast diffusion in the liquid; (iii) small enough radius

Impact of truncated edge on crystal structure

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:

When main facet does not meet the trijunction, islands do not nucleate at the trijunction!