www.sciencemag.org/cgi/content/full/335/6074/1330/DC1

Supporting Online Material for

Scaling Hetero-Epitaxy from Layers to Three-Dimensional Crystals Claudiu V. Falub,* Hans von Känel, Fabio Isa, Roberto Bergamaschini, Anna Marzegalli,

Daniel Chrastina, Giovanni Isella, Elisabeth Müller, Philippe Niedermann, Leo Miglio

*To whom correspondence should be addressed. E-mail: cfalub@phys.ethz.ch

Published 16 March 2012, Science 335, 1330 (2012) DOI: 10.1126/science.1217666

This PDF file includes:

Materials and Methods SOM Text Figs. S1 to S12 References (32–36)

2

Materials and Methods

Substrate patterning Most of the 100 mm Si(001) substrates were patterned by conventional photolithography and deep reactive ion etching (DRIE) based on the Bosch process (32). This technique is widely used in microsystems fabrication. It provides very good anisotropy and high etch rate (~ several micrometers per minute) but rough sidewalls. We used patterns with uniformly spaced pillars, as well as pillars arranged in 10×10 blocks separated by wider trenches. In order to explore the influence of sidewall roughness on the epitaxial growth, a cryogenic process (33) was used for a number of substrates yielding smooth sidewalls. In addition to the usual patterns with clean pillar sidewalls and trench bottoms, some patterns received a passivating oxide coating (~90 nm SiO2), leaving only the top of the pillars oxide-free for epitaxial growth. Crystal growth Samples were grown by low-energy plasma-enhanced chemical vapor deposition (LEPECVD), a process based on an Ar plasma sustained by a low-voltage, magnetically confined direct-current arc discharge between a heated Ta filament and a ring anode in electrical contact with the grounded chamber walls. The potential distribution between the filament and the anode gives some directionality to the ionic species in the plasma, which in turn gives rise to the vertical component of the flux used in the modeling (27, 34). Pure germane (GeH4) is used as the reactive gas, while silane (SiH4) can also be added to deposit SiGe alloys of any composition. The base pressure in the load-locked growth chamber was below 110-9 mbar, whereas during growth the pressure was ~210-2 mbar. The nominal growth temperature, TG, was measured with thermocouples located behind the heater, and the correspondence between these temperature readings and the true substrate temperature was calibrated by means of an infrared pyrometer. However, since the substrate is directly exposed to a high-density plasma during deposition, the actual surface temperature is difficult to determine, and may differ from the calibration obtained by optical pyrometry in UHV. Prior to deposition, the patterned Si substrates were cleaned using the industry standard RCA method (35), and the native oxide was removed by a dilute (5%) HF dip and subsequent rinse in ultrapure water. Once loaded into the growth chamber, the substrates were outgassed in UHV for 15 min at 300C before ramping to the growth temperature. Analytical methods to investigate the surface morphology, dislocation density and crystalline quality of Ge towers The morphology of the Ge towers grown on the patterned Si substrates was monitored by Nomarski interference contrast optical microscopy (Nikon Eclipse 200D) and scanning electron microscopy (SEM) (Zeiss ULTRA 55 digital field emission). Facet orientation was accurately determined by atomic force microscopy (AFM) using a Veeco Innova microscope, and transmission and scanning electron microscopy (TEM, STEM) using a Tecnai F30ST TEM/STEM transmission electron microscope (FEI), operated at 300 kV. The specimens for TEM/STEM investigations were thinned to electron transparency by mechanical thinning followed by Ar-ion milling (acceleration voltage 4.3 kV, etching angle 4°, final polishing at 1 kV acceleration voltage).

3

Threading dislocations in Ge towers were inspected by TEM. The dark field STEM technique was used (in [110] zone axis orientation) to improve the visibility of dislocations in areas of high dislocation density by suppressing the contributions of bending contours and thickness fringes to the image contrast. Defect etching was used to estimate the dislocation density. The Ge towers were etched for 40 seconds in a diluted iodine solution (15 mg I2, 33 ml glacial acetic acid, 10 ml 65% nitric acid, 5 ml 40% HF, etch rate about ~10 nm/s) at 0°C. Etch pits were counted by AFM.

High-resolution x-ray diffraction (HRXRD) was used to investigate the crystalline quality and strain of the Ge towers. The HRXRD measurements, e.g. rocking curves, -2 scans and reciprocal space maps (RSMs), were performed with Cu K1 radiation using a PANalytical X’Pert Pro-MRD diffractometer equipped with a 4-bounce Ge(220) crystal monochromator on the incident beam, and an analyzer crystal and a Xe point detector on the diffracted beam. The x-ray beam diameter on the sample was ~1 mm. Therefore, depending on the lateral pattern dimensions, several thousands of Ge towers were probed simultaneously. To assess the tilt of individual Ge towers submicrometer diffraction experiments were performed at the ID01 beamline of European Synchrotron Radiation Facility (ESRF) in Grenoble, using an 11.07 keV x-ray beam focused down to ~300500 nm by means of Fresnel zone plates (FZP) and a Huber diffractometer equipped with a high-precision (x,y,z) stage. For a certain reflection, e.g. symmetric or asymmetric, and a fixed (x,y) position, the incidence angle of the x-ray beam was scanned while moving the beam across the sample. Since a 2D pixel detector was used, three-dimensional RSMs could be constructed for each (x,y) position of the x-ray beam on the sample.

Supporting Text Estimation of the separation of Ge towers Low resolution STEM micrographs were used to determine quantitatively the distance between the Ge towers as a function of their height. For ~7-μm-tall Ge towers grown at 440°C on an array of 8-μm-tall and 2-μm-wide Si pillars spaced by 4 μm trenches the result is presented in fig. S1. We see that after ~2 μm Ge deposition the lateral growth has become negligible, and, except for the top region exhibiting facets, the separation between the Ge towers remains constant at ~55 nm. A very similar self-limitation has been observed for much thinner deposits on top of narrowly spaced Si pillars (or mesas), as discussed below in conjunction with fig. S5. Whether the effect of full lateral growth suppression below a critical distance can be attributed to the low sticking probability of molecules on the {110} facets below a critical grazing angle of the incoming flux, or to the efficient downward diffusion of deposited material towards the trench bottom is not yet clear, and more detailed investigations are required. Ubiquity of epitaxial tower morphology The space-filling arrays of epitaxial crystals obtained by self-limited lateral growth on top of Si pillars were found to be remarkably independent of any detail of the fabrication procedure, as long as the basic conditions outlined in the text were fulfilled.

Fig. S8-µm-were 55 nm

Fig. Spillar(A) is

a) GThe owidthtrenchphenoGe tofig. Sachie

A

A

S1. (A) Dark-tall and 2-µused to dete

m.

S2. Top view rs with spacin 8 µm, where

Geometry ofonly effect th after their hes at the oomenon of owers grownS2. As the eeved for muc

k field STEM µm-wide Si permine the he

SEM microgngs of: (A) 1eas for (B-E)

f the substrathe spacing lateral expa

onset of the self-limitedn onto 2-μmexample of ch thinner s

B

micrograph pillars spacedeight depend

graphs of Ge 1 µm; (B) 2 µ it is 27 µm.

ate patternof the substansion has sgrowth the

d lateral expm-wide Si p200-nm-wi

structures if

C

of ~7-μm-tad by 2 µm tr

dence of the

towers growµm; (C) 3 µm

trate pillars stopped. Obthicker the ansion. This

pillars with ide trenchesthe trench w

C

ll Ge towers renches. (B)inter-tower d

wn at 490°C om; (D) 4 µm

has on the fbviously, hodeposit nees can be seespacings ra

s of fig. S5width is furt

B

D

grown at 44The yellow

distance D, c

onto 2-µm-wm; (E) 5 µm.

final Ge towwever, the w

eds to be in en from the anging betw

shows, simther reduced

40°C on an acontour lineconverging t

wide and 8-µmThe Ge cove

wers concernwider the suorder to revtop SEM v

ween 1 and 5milar results d.

E

4

array of

es in (A) to about

m-tall Si erage in

ns their ubstrate veal the

views of 5 μm in can be

Fig. S[110]) (b) EThe intentexamaxes directnorm

(c) IThe sevolugrowpronosidewpolyc

(d) SA furSEMby 201.7 μsimil

(e) TA moexpanmism

S3. SEM mic), patterned S

Effect of suboverall towtional miso

mple of Ge toof the towtion, as may

mal.

Initial facet sidewall strution of the

wn, under oounced rippwalls (fig. Scrystalline g

Submicromerther manife

M images of 00-nm-wideμm. Accordiar to those o

Thermal andost strikingnsion of epi

match. To thi

crographs ofSi wafers wit

bstrate misower morphoorientation owers grow

wers are eviy be seen fr

distributionructure of the towers. Thotherwise idples caused S4). Evidentgrowth befor

eter trenchesestation of tfig. S5, sho

e trenches. Tng to the inobtained for

d lattice misg example oitaxial crystais end, we c

f ~8-µm-tall th 8-µm-tall

orientation ology doesof the Si s

wn onto a Si(idently defiom the top

n he Si pillarshis follows dentical con

by the Botly, some Gre flux shiel

s and smootthe unimpo

owing 500 nThese trenchnset of fig. r wider trenc

sfit of the geneals has been

compared ep

Ge towers gand 2-μm-wi

s not appeasubstrate. T(001) substrined by the(001) facet

s turned outmost drast

nditions, onosch procesGe reached lding by nei

th pillar sidortance of thnm Ge depohes were etcS5A, the geches.

eral validityn found by lpitaxial depo

grown at 44ide pillars, sp

ar to depeThis followrate misoriee growth diinclined by

t to be irreltically fromn Si pillars, and on Sthe oxidize

ighboring to

dewalls he sidewall sits on diffe

ched by cryoeneral shape

y of space looking at thosits of pure

0°C on misopaced by 2 μm

end on anyws from figented by 6° tirection rathy 6° with res

levant in term a comparirs with cleaSi pillars wed sidewallsowers set in.

structure caerent sized Sogenic DRIEe of Ge tow

filling by she effect of e Si, Si0.6Ge

oriented (6° m trenches.

y unintentiog. S3, showtowards [11her than thespect to the

rms of the ison of Ge an sidewall

with oxide-cs, leading to.

an be foundSi mesas seE to a depth

wers is indee

self-limited lattice and t

e0.4 alloys an

5

towards

onal or wing an

0]. The e [001] surface

general towers

ls with covered o some

d in the parated

h of just ed very

lateral thermal nd pure

Fig. Sµm-tapillar Ge, adjusS6, intower

Fig. S~1.7-µpillar(DRIEbetwe

A

S4. SEM micall and 2-µmrs with passiv

obtained usted to yieldndicates thar morpholog

S5. Perspectiµm-tall Si p

rs were fabricE). (A) 0.8-µeen two Ge to

crographs of -wide pillars

vated (~90-nm

under identid approximaat neither thgy in any sig

ive SEM micpillars with dcated by elecµm-wide Si powers); (B) 1

A

~8-µm-tall Gs, spaced by m-thick SiO2)

ical conditiately similarhe lattice mignificant wa

crographs of different aspctron beam lipillars (inset1.3-µm-wide p

B

Ge towers gr2 μm trenche) sidewalls.

ions, excepr surface diismatch noray.

f ~500-nm-tapect ratios sethography (Et shows a mpillars; (C) 1

B

rown at 560°es: (A) Si pil

pt for diffeffusion leng

r the therma

all self-limitedeparated byEBL) and cryagnified cros1.8-µm-wide

C on patternllars with cle

erent substrgths. The real mismatch

d Ge towers ~0.2-µm-wi

yogenic deepss-section SE

e pillars.

C

ned Si wafersean sidewalls

rate temperesult, shownh affect the

grown at 46ide trenches. reactive ionEM view of

6

s with 8- s; (B) Si

ratures, n in fig. general

60°C on The Si etching the gap

Fig. Slimiteμm trpure G

Fig. SXRD onto am-won thcorrestowerGe filscansunpatcrack

A

S6. Perspected lateral grorenches: (A) pGe towers gr

S7. Analysis measuremena Si substrat

wide trenchese unpatternesponding pers at the outelm. (C) Persps along the Qtterned regioked, continuo

tive SEM miowth on pattepure Si towe

rown at 485°

of crystal qunts around thte patterned

s. Individual ed part of theaks originatermost rows opective view Qz direction ions, and a hus Ge film an

icrographs oerned Si subsers grown atC.

uality and strhe Ge(004) r

into 1010 blocks are s

he substrate.te: I – relaxof blocks; IIISEM microgin the vicinityhigh-quality nalyzed in (B

B

of ~8-µm-talstrates with 8t 810°C; (B)

rain of self-lireflection forblocks of 2-eparated by Roman lett

xed symmetrI – strained Ggraph of the Gty of Ge(004)

Ge wafer. B) and (D).

ll three-dime8-µm-tall and Si0.6Ge0.4 al

imited Ge towr: (A) 16-mm-wide and2.5-m-wide

ters indicate ric Ge towerGe between tGe towers an) Bragg refle(E) Cross s

C

ensional crysd 2-µm-widelloy towers g

wers on Si(0m-tall Ge tow

d 8-m-tall pe trenches. (B

the spatial rs; II – relathe rows of pnalyzed in (Aection, compsection SEM

C

stals grown e pillars, spacgrown at 660

01). High res

wers grown apillars spaceB) Continuouregions from

axed asymmepillars; IV – sA) and (D). (Dparing pattern

M micrograph

7

by self-

ced by 2 0°C; (C)

solution at 440C ed by 1-us layer

m which etric Ge strained D) x-ray ned and h of the

Fig. SSi(224reflecreciprthe Siµm-wthe G AnalyThe cdeterarounfor threflecarountwo con tohighebetweclearlwas ~in the

S8. Analysis4) reflection

ction in (A). Nrocal lattice i(004) reflect

wide and 8-µmGe(004) reflec

ysis of straiconcentratiormined fromnd the Si(00he small (lection with thnd the surfacomponentsop of the Si per Qz momeen the Si ply observed~33°, but ine (224) map

s of strain rs for the saNote that theorigin. Peakstion for 8-µmm-tall pillarsction in (C).

n relaxationon, degree o

m the HRXR04) and Si(2ess than 0.5he [110] an

ace normal).s, an intensepillar (peak

mentum valpillars (peakd in the (00n view of theps (fig. S8B)

relaxation of ample presene elongation ks indicated bm-tall Ge tows spaced unif

n and tilt of of strain relRD measure224) Bragg p5°) substratend [110] dire

Basically, te componentk I in figs. S7lues, stemmk III in figs. 4) maps (fie attenuatio) for which t

f Ge towers nted in Fig.

of the Ge(22by Roman letwers grown aformly by 1-µ

Ge towersaxation, and

ements perfopeaks (figs. e miscut byections in ththe diffractit stemming 7 and S8), a

ming from S7 and S8)

igs. S7A ann of x-rays the incidenc

on Si(001). 4A. (B) Det24) reflectiontters are defint 490C ontoµm-wide tren

d the tilt of formed with

S7 and S8)y performinghe diffractioion peaks frfrom the fu

and a much the tensile

). For 16-µmnd S8D) for

in the tall toce angle ω w

(A) RSMs atailed view an occurs tangned in fig. S7o a Si substranches. (D) D

f Ge and Si1

h the laborat). These valug two meason plane (i.erom the Ge ully relaxed

weaker come-strained mm-tall Ge to

which the owers it couwas only ~6

around Si(00around the Ggentially aro7A. (C) RSMate patterned

Detailed view

1-xGex towertory diffractues were cosurements foe. a rotationor Si1-xGex material de

mponent, shmaterial deowers, peak

incidence auld not be re6.6°.

8

04) and Ge(224) ound the

M around d into 2-

w around

rs were tometer orrected for each n of 90°

exhibit eposited hifted to eposited III was

angle ω esolved

Fig. SPersppatterSchemray b[110]In viepassepillarcorres The tnumbwas ebroadThe tdeterfocusS9B)

S9. Synchrotpective view rned into 2-matics of the eam was foc

] direction. Aew of the shaes through sers. (D,E) Psponding to t

tilt angle ofber of towerestimated frdening (i.e. tilt of an in

rmined fromsed (~3005).

tron submicroSEM microg

-m-wide ansubmicrome

cused by FreA two-dimensallow incideneveral Si pillaPosition sens

the sample in

f the Ge towrs probed brom the diff

full-width ndividual Gm the synchr500 nm) x-

ometer diffragraph of 1-µ

nd 8-m-tall eter difractioesnel zone plsional (2D) pce angle for ars at differesitive XRD n (A).

wers with reby the 1-mmfuse scatterinat the half e tower, whrotron submray beam th

action experiµm-tall Ge tl pillars spaon experimenates (FZP) t

pixel detectorSi(004) and

ent heights wintensity m

espect to them-wide x-rang of the Gmaximum,

hich is causmicrometer dhat was sm

iments perfotowers grow

aced uniformnts for Si(004to ~300500 r was used toGe(004) refl

when the diffrmaps for Si(

e [001] direay beam of

Ge(004) diffrFWHM) a

sed by its ddiffraction e

mall enough

rmed at the n at 490°C

mly by 1-µm4) and Ge(00

nm and waso record the dflections, the fracting plane(004) and

ction, averathe laborato

raction peakalong the Qdistinct netwexperimentsto probe a

ID01 beamlonto a Si su-wide trench

04) reflections oriented aldiffracted beincident x-ra

e intersects aGe(004) ref

aged over thory diffractk by measur

Qx momentumwork of MDs performed single tow

9

line: (A) ubstrate hes. (B) n. The x-long the eam. (C) ay beam a row of flections

he large ometer, ring the m axis.

Ds, was d with a wer (fig.

In or490°Cfirst aunpatat difcovercan arelatifor Gone Son thacrosbelowobserexhibpillarGe(0a ranto the FiniteIn ordthe rCOM

Fig. Sand 8and relaxaµm-w(red a

rder to visuaC on 2-µm-aligned the tterned regifferent (x,y)rage (1 µm)assume that ively shallow

Ge(004)). HoSi pillar willhe sample (dss different pw and Ge torve that thebits maximars being par04) reflectio

ndom mannee Si pillars.

e element sider to interprelaxation o

MSOL Multi

S10. (A,B) FE8-µm-tall Si pzz strain comation of therm

wide and 8-µmarrows) was a

alize this ti-wide and 8sample (i.e.ion, and then) positions i), and of theonly a singlw angle of owever, duel contribute due to the shpillars at difowers abovee (x,y) posa arranged irt of the subon exhibits er (fig. S9E)

imulations pret the XRof thermal iphysics soft

FEM simulatipillar. The remponents on mal strain (m-tall Si pillamplified by

lt for a sam8-µm-tall pi surface norn we measun the pattere relatively le tower conincidence (f

e to the peneto the Si(00

hallow incidfferent heige were nicelsition sensitin the regul

bstrate (fig. maxima tha). This prov

RD measuremstrain in G

ftware packa

ons of thermed and blue c

z along the zz strain comlar. The displa factor of 4

mple consistillars spacedrmal) for theured the interned area (filarge separa

ntributes to for 11.07 keetration dept04) signal. Tdence angle

ghts. Neverthly resolved tive XRD lar pattern S9D). On that are slightves that the

ments, FinitGe towers age.

mal strain relacurves in thecenter of th

mponent) in alacement fiel40 to highligh

ting of 1-µmd by 1-µm-e Si(004) reensity of Siig. S9, D anation betwethe Ge(004eV this is 24th of the incThis, togethee), will meanheless, we o(fig. S9, D intensity mof the undehe other hantly displacedGe towers

te Element Mwere perfo

axation in a adjacent gre tower (x =

an asymmetrild indicated bht the tilt with

m-tall Ge to-wide trencheflection of t(004) and G

nd E). In vieeen the Si pi) signal (fig4.359° for Scoming x-raer with the wn that informobserve that

and E). Onmap of the erlying subsnd, the corred from the Sare slightly

Method (FEormed using

Ge tower onaphs show th

= 0). (C) FEic Ge tower gby the three-h respect to t

owers depohes (fig. S9the substrat

Ge(004) reflew of the smillars of 1 µ

g. S9C), despSi(004) anday beam, mowider beammation is avt both the Sin the one haSi(004) ref

strate, owinesponding mSi(004) maxtilted with

EM) simulatg the comm

n top of a 2-µhe dependenc

EM simulatiogrown on top-dimensionalthe [001] dire

10

sited at 9A), we te in the lections mall Ge µm, one pite the

d 23.32° ore than

m profile veraged i pillars and, we flection

ng to Si map for xima in respect

tions of mercial

µm-wide ce of xx n of the p of a 2- l vectors rection.

11

For symmetric Ge towers grown on patterned Si substrates with constant spacing between the Si pillars, the simulated three-dimensional geometry consisted of a Si cylinder (2 µm in diameter and 8 µm in height) and a quadratic prism with a truncated pyramidal cap Ge tower on top (as tall as 8 µm), connected by a downward truncated pyramidal part, surrounding the Si cylinder. The numerical calculations were performed by meshing this geometry with 185200 elements and by considering an initial hydrostatic expansion in the Ge pillar equal to 0.20%, as derived from the difference in the thermal expansion coefficient of Si and Ge for a step in temperature ~500°C. As detailed in fig. S10, A and B, the strain components along a central axis of the Ge towers approach zero a few hundred nanometers above the Si/Ge interface.

To qualitatively confirm the tilt of the asymmetric Ge towers at the edge of the 10×10 blocks observed in the HRXRD RSM measurements (figs. S7 and S8), an asymmetric structure was analyzed by FEM simulations (see fig. S10C, where the map of the zz strain component is reported). Here we display also a magnified displacement pattern (red arrows), indicating that an actual tilting of the tower is qualitatively predicted. Additional remarks on the growth model For the two-dimensional simulations of the morphological evolution with deposition, we considered the following rate equations of the adatom density N for each small segment of a facet:

ΦC D

C ,

where is the site-dependent incident flux, C⁄ is the facet-dependent rate of crystallization (that is, the facet growth velocity ), and D⁄ is the rate of desorption/etching of adatoms from the surface toward the gaseous phase. Actually, C and

D are the respective adatom lifetimes, the former depending on the facet, the latter taken to be the same for each facet.

From the experimental SEM profiles for an isolated pillar at fixed growth temperature, starting from ~1 μm Ge deposition, we observe that the growth almost preserves the ratios between the growth velocities of the different facets. This is particularly evident in the linear trajectories of the vertices between the facets (dashed white lines in Fig. 3C). We can consider the system to be in a stationary state, and use the simplified equations reported below. This also means disregarding the process of initial facet nucleation, which cannot be treated by the model, but turns out to be irrelevant for the experimental tower morphology, as discussed in a previous section.

Under stationary conditions, the rate equations for each segment can therefore be written as:

0 ΦC D

Φ

C CΦ ∙ Φ,

Fig. Scollecmutua

wherinto fnecesadatoand istatioof a m

In ordrespeprogrand vneighcase, base {110S11B~150the mexpla

S11. Schemact a lower flal shielding o

re 1facet {hkl}.ssary in theom phase dein turn Φ00

onary value monolayer f

der to betterect to the ressively stevertical Φ2 hboring crys

the smoothis entirely d} sidewalls

B). The resu0 nm. The lamore pronoains the obs

atic drawing flux (both in of close neigh

C/ D-1 is

If we neede model), wensity N sm

01=4.4 nm/sfor N is rea

for any facet

r explain thmaximum

eeper facetscomponent

stals furtherh decrease odue to the di, in turn ge

ult is a steparger the graounced becoerved sidew

showing: (Aisotropic

hboring crys

the factor dd a referenc

we arbitrarilymaller than o

s, to matchached after t.

he reductionvalue Φ00

s of an isolats are both r reduces th

of the simulaifferent growenerated byped {110}

adient in theomes the m

wall profile

A) progressiv1 and vertic

tals further r

describing tce value for y set C=0.

one monolayh the experdepositing

n of the inco01, we schated Ge crys reduced),

he isotropic ated width owth velocity

y the non-unprofile in w

e flux along macroscopic(see Fig. 3D

vely steeper fal 2 compo

reduces the is

the fraction on a plan

01 s and Dyer. This edrimental groa few nm o

oming flux thematically stal collect and how thcomponent

of closely spy of neighboniform inco

which each the sidewal

c curvatureD and fig. S

facets of an onents definsotropic comp

of incidentnar (001) suD=0.1 s, in

ducated guesowth rate

of Ge, stayin

to an inclinedisplay in

a lower fluxhe mutual t Φ1 (fig. Spaced Ge tooring segmeoming flux plateau has

lls, the tallere of the prS1). In the in

isolated Geed in Fig. 3

mponent of the

t flux incorpurface (not order to h

ss gives 00

=4 nm/ng within fr

ed facet {hkn fig. S11Ax (the isotroshielding o11B). In thi

owers towardents compos(see insets

s the dimenr are the sterofile. This ntermediate

12

e crystal 3A); (B) e flux.

porated strictly ave the

01=0.91, /s. The ractions

kl} with A how opic Φ1

of close is latter ds their sing the

in fig. nsion of eps, and

nicely e region

of thincomtowerpart oincomIn oustopsare jmorp

For tdiffusdrivebetweshortthen b

in whfacet-s coo

Fig. Sdiffus

he tower sidming flux surs was criticof the towerming flux wur model wes the lateral just the re

phology.

the higher gsion betwee

en by N gradeen facets. er, or at bebe re-writte

ΦC

hich the las-dependent

ordinate alon

S12. Compasion length of

dewall (reguffered the cally shrinkr sidewall (r

was uniform e did not incgrowth at so

esult of the

growth tempen facets isdients alongThe former

est comparaben as (25):

D

C ,

st term in tdiffusion co

ng the surfac

arison betwef: (A) 100 nm

ion 2 in figlargest vari

king. In the region 3 in and close toclude any mome 50 nm e flux redu

peratures uss likely no g the surfacr kinetic effble, to the

the first eqoefficient, ace profile in

een the simum; (B) 1 μm.

g. S11B), tiation durinlower part fig. S11B)

o zero by fumicroscopic

distance beuction, stil

sed in somelonger negle profile, or

fect acts on facet dimen

quation reprand the secon our two-di

ulation profil

the profile ng tower gro

(region 1 inthe (110) fa

ull shieldingmechanism

etween Ge cl providing

e of the expligible, andr by differenthe scale o

nsions. The

esents diffuond derivativimensional s

les for diffe

is rather cuowth, as then fig. S11Bacet is nearl.

m (still not dcrystals: ourg the essen

periments (ed a net diffunces in the f diffusion system of

usion by Five is expresscheme.

erent growth

urved becaue distance b

B) and in thely flat, beca

determined)r simulated pntial experi

e.g. Fig. 4Cfusion flow

chemical plengths, whrate equatio

ick's law, Dssed in term

stages assu

13

use the between e upper

ause the

, which profiles imental

C), local can be otential

hich are ons can

D is the s of the

uming a

14

Our results for 2-µm-wide Si pillars, spaced 4 µm apart, are displayed in fig. S12A, using kinetic parameters extracted from the experimental evolution of an isolated tower and a diffusion length of the order of 100 nm (essentially no diffusion). By considering now the diffusion length inside each facet to be of order of 1 µm, comparable to the facet dimensions (still artificially keeping the same incorporation factors as for the previous case), we obtain both a retarded lateral growth and an expanding {111} top facet (fig. S12B). The growth of {111} facets at the expense of {113} facets is not, however, consistent with the experiments showing the opposite trend (Fig. 4C), so we conclude that even at the higher growth temperature diffusion is limited to a few hundred nanometers. Still, the experimental distribution of Ge material at the bottom of the trenches (Fig. 3D), seems to be smoother than our simulations with no diffusion and more consistent with the simulations based on a micrometer-sized diffusion length (fig. S12B). This observation supports the argument that by lowering the incoming flux (as happens at the bottom of the trenches) the diffusion is increased to common values expected for low rate depositions. Having explained the observed tower morphology in terms of kinetic (independent facet growth by short diffusion lengths) and geometric (mutual flux shielding) features, the question arises as to what extent the results remain valid for growth techniques other than LEPECVD. Clearly, some degree of flux shielding is always expected since in reality an in-plane flux component is present even for rather collimated particle beams, such as those in MBE. Still, since growth rates in MBE are generally much lower, either a surfactant or much lower substrate temperatures would have to be used, in order to guarantee short surface diffusion lengths (36).

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