development of growth theory for vapor–liquid–solid

Research ArticleDevelopment of Growth Theory for VaporndashLiquidndashSolidNanowires Wetting Scenario Front Curvature Growth AngleLinear Tension and Radial Instability

Valery A Nebolrsquosin Nada Swaikat and Alexander Yu Vorobrsquoev

Voronezh State Technical University Department of Radio Engineering and Electronics Voronezh 394026 Russia

Correspondence should be addressed to Nada Swaikat nadas84mailru

Received 25 October 2019 Accepted 27 February 2020 Published 28 March 2020

Academic Editor Marco Rossi

Copyright copy 2020 Valery A Nebolrsquosin et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper we report that under wetting conditions (or modes) of nanowire (NW) growth when a nonplanar crystallizationfront emerges under a catalyst droplet a shift in the three-phase line (TPL) of the vaporndashliquidndashcrystal interface occurs underthermodynamically stable conditions when the angle with respect to the droplet surface termed the growth angle is fixed egrowth angle of the NWs is determined not from a geometrical perspective but on the basis of the physical aspects of the processesoccurring around the TPL revealing a size dependence caused by the influence of linear tension of the three-phase contact of avaporndashliquid crystal e observed radial periodic instability of the NWs is described according to the size dependence of thethermodynamic growth angle which induces negative feedback in the system Under the influence of linear tension and positivefeedback the tips or needles of NWs can be formed

1 Introduction

e processes of nanowire (NW) growth in semiconductingmaterials through a vaporndashliquidndashcrystal (VLS) mechanismdescribed in most studies including [1ndash11] are based on theestablished role of the free surface energy of the interfacesand on the flat border of the crystallization front under acatalyst droplet (nonwetting condition (or mode) of whiskergrowth [1 2 4 5 9 10]) In principle however this ap-proach is incorrect because an analysis of the free surfaceenergy can be justified only when one phase is completelydispersed within the volume of the other phase e growthof NWs is a well-known example of crystallization in whichthree phases simultaneously participate in equilibrium in amulticomponent systemematerial of each phase near theclosed line of the three-phase contact is under the influenceof the materials of the two other phases thus its state differsfrom the state within the volume and from the state near theinterface of the other two phases [12] e presence of athree-phase border is the most pronounced feature of NW

growth Hence the quasi-one-dimensional growth ofnanosized crystals cannot be correctly described withoutconsidering the free energy corresponding to a lineartension of the curved interface of the three phases Here it isnecessary to mention that the effects of the surface curvatureof the NWs and the catalyst droplet on the surface energyshould appear at radii of the order of unity and one-tenth ofa nanometer which corresponds to the intermoleculardistance In the case of linear tension the effect of thecurvature of the three-phase border along the wetting pe-rimeter of the droplet occurs at radii of tens of nanometerswhich corresponds to the radius of action of the surfaceforces

During the VLS growth of the NWs with the smallcontact angles of the catalyst droplets at the top of the NWsa three-phase VLS border along the wetting perimeter of thedroplet adjoins the singular face of the crystallization frontIn such a case a condition similar to a melt pinned to acrystal edge occurs when the methods developed by Czo-chralski and Stepanov are used ie when the catalyst droplet

HindawiJournal of NanotechnologyVolume 2020 Article ID 5251823 9 pageshttpsdoiorg10115520205251823

wets the facet of the crystallization front and does not wet asidewall surface of the NWs (nonwetting condition (ormode) of NW growth [10 11 13]) For a nonwetting growthmodel the linear energy that occurs when the catalystdroplet does not wet the sidewall surface of the NWs and aflat crystallization front appears under the droplet wascarefully analyzed in [7 14 15] However it is well knownthat at large contact angles of catalyst droplets on the top ofthe NWs at the crystallization front curved sections andeven oblique densely packed faces can appear leading to theformation of facets on the sidewall of the crystal [3 16] Anonplanar geometry of a crystallization front was realized inthe wetting model of NW growth [10 11] Nevertheless inthe literature the linear energy of a three-phase borderunder wetting conditions has not been described

In the present article we explain that under the wettingconditions for steady NW growth (without considering thekinetic effects) a shift in the three-phase line (TPL) of theinterface boundaries occurs at a fixed angle with the surfaceof the catalyst droplet Additionally we reveal the size de-pendence of the thermodynamic growth angle of the NWsdetermined according to the influence of the linear tensionof the three-phase contact vaporndashliquid crystal We alsoshow that the growth angle of the NWs should be distin-guished from the wetting angle which characterizes the localequilibrium toward the movement of the liquid droplet overthe surface of a solid body and is not directly related tocrystallization Furthermore we demonstrate that the ob-served radial periodic instability of the NWs can beexplained on the basis of the size dependence of the ther-modynamic growth angle which induces a negative feed-back mechanism in the system Tip or needle NWs can beformed under the influence of linear tension and the actionof a positive feedback mechanism

2 Wetting Model of NW Growth

e cross-sectional dimensions of NWs are usually designedaccording to the linear dependence on the size of the catalystdroplet As a rule the ratio of the wire diameter to thedroplet diameter is sim(07 09) [13 17 18] We assume thatduring the growth of cylindrical NWs the border of thethree-phase contact takes the form of a circle is corre-sponds to the thermodynamically stable form of a TPLwhich is the shortest length for a given area of a crys-talndashliquid interface

Assume that a liquid catalyst droplet with a certainvolume is placed on top of the NWs with a composition inaccordance with the phase diagram of a metal-ndashsemiconductor (Figure 1) As the geometrical parametersof the droplet we choose the base radius r and angles be-tween the tangent and the meniscus of the droplet of theliquid passing through some point A along the TPL withhorizontal θ and vertical φ (the NWs are grown in thevertical direction) Suppose that in the wetting regime [10]under the droplet a stepped crystallization front with athickness of z is formed Over time the TPL shifts in the z-direction according to the angle δ between the vertical andthe tangent or the angle ψ between the horizontal and the

tangent to the surface of NWs growing at point A (where ψis the inclination angle of the curved surface of the crys-tallization front)

en the shift of l from pointA to pointAprime is determinedas follows

l z

cos δ (1)

A shift in the TPL on l is accompanied by a reduction ofthe crystalndashliquid interface area by 2πrl sin δ an increase inthe surface crystal area by 2πrl a decrease in the liquidsurface area by 2πrlcos(φ+ δ) and a decrease in the length ofthe TPL by 2πrl sin δ Hence the change in the surface (ΔFS)and the linear component of the Helmholtz free energy(ΔFl) which is associated with the indicated shift of point Acan be expressed as follows

ΔF ΔFs + ΔFl 2πrlαS minus 2πrlαL cos(φ + δ)

minus 2πrlαSL sin δ minus 2πlχ sin δ(2)

where αS αL and αSL represent the free surface energies ofthe vaporndashcrystal vaporndashliquid and crystalndashliquid inter-faces respectively and χ represents the tension of a closedline of the three-phase VLS contact

In equation (2) the first term represents the increment inthe free surface energy of the crystal due to an increase in itssurface area through a variation in the droplet to height dzthe second term represents a decrease in the free surfaceenergy of the liquidndashvapor interface the third term repre-sents a decrease in the free surface energy of the crys-talndashliquid interface and the fourth term represents adecrease in the extra linear energy due to a reduction in theTPL length

By substituting (1) into (2) then differentiating (2) withrespect to angle δ and equating the derivative to zero(d(ΔF)dδ|δδlowast 0) we determine the shift in the TPLcorresponding to theminimum increment in the free surfaceand linear energies as follows

αS sin δlowast

minus αL sinφ + αSL +χr (3)

In equation (3) χr represents the two-dimensionalpressure αχ occurring from the bend of the contact line(analogous to the capillary Laplace pressure arising from thecurvature of the interface surface) [12] and 1r representsthe curvature of the line e absolute value of the lineartension χ is estimated to be between 1times 10minus 11 and 1times 10minus 9 Jm[7 19]

Equation (3) indicates that the angles δlowast and φ alongwith the free surface energies αS αL and αSL and the linearenergy αχ determine the equilibrium condition of thedroplet at each point of the TPL on top of the whisker

Equation (3) implies an equilibrium condition of thedroplet of a liquid lying on a horizontal solid surface (atδlowast 90deg)

αS αL cos θ0 + αSL +χr (4)

We obtained modified Youngrsquos equation for the me-chanical equilibrium of a droplet of liquid on a flat surface

2 Journal of Nanotechnology

taking into account the contribution of the linear tension χ[20 21] Here θ0 represents the contact angle (Figure 2(a))

At δlowast 0 a shift in the TPL occurs in the vertical di-rection (along the axis of the NWs) and the equilibrium ofthe droplet is thus determined as follows

αL sinφ αSL +χr (5)

which coincides with a similar equation obtained for thenonwetting growth of cylindrical NWs [10]

For a system of Au-Si where r 1times 10minus 7m αL 091 Jm2αSL 045 Jm2 and two angles φ are 32deg and 35deg [22] usingequation (5) we calculate the linear tension χ and linearenergy αχ For φ 32deg we obtain χ 322times10minus 9 Jm andαχ 322times10minus 11 Jm2 us αχ is approximately 7 of thefree interfacial energy of the crystalndashliquid interface Forφ 35deg we obtain χ 720times10minus 9 Jm Accordinglyαχ 720times10minus 11 Jm2 which is sim16 of the set value of thefree surface energy αSL

3 Growth Angle of NWs

When 0le δlowast le 90deg the configuration of the small catalystdroplet on top of the NWs must strictly correspond to theangles θ and ψ e difference between them (θ minus ψ φ+ δlowast)is the thermodynamic growth angle of the NWs φ0 that isthe angle between the tangent drawn to the meniscus of theliquid phase at the selected point at the interface of the threephases and the direction of the equilibrium shift of the TPL

[23 24] e increment in the free energy of the three-phasesystem at the shift of the TPL under φ0 is zero further theformation of the NW surface under other angles differingfrom φ0 leads to an increase in the free energy during thisprocess and therefore cannot occur Hence φ0 is theequilibrium angle of the TPL shift during the growth of theNWs e tendency of the NWs to grow with a constantdiameter is determined by the fact that during the growth ofthe cylindrical crystals the increment in the areas of theinterfaces is minimal

e phase of zincblende GaAs NWs is formed at small(lt100deg) and large (gt125deg) contact angles θ and accordinglyat φ0lt 10deg and φ0gt 125deg whereas the pure wurtzite phase isobserved for intermediate contact angles [25]

To obtain an equation for the growth angle of the NWsφ0 in an explicit form we first apply the axial and radialcomponents of the projections of the force vectors corre-sponding to the free surface energies of interfaces αS and αSLat point A on the TPL for large angles of macroscopicdroplets φ0⟶φinfin0 (without taking into account the con-tributions of the linear tension and the anisotropy of the freesurface energy at the interfaces) is yields the followingsystem of equations

minus αS + αL cosφinfin0 + αSL sinψ 0

minus αL sinφinfin0 + αSL cosψ 01113896 (6)

where φinfin0 represents the angle between the tangent drawn tothe meniscus of the liquid phase at the selected point A at the

S

zl

LV

R

A

2r

Aprime

φ0

αSLαχψ

θδ αS

αL

φ

(a)

S

zl

LV

R

2r

A

AprimeαSL αχ

αS

αL

φ = φ0

(b)

Figure 1 Wetting model of VLS growth of NWs (a) conical (δgt 0) and (b) constant radius (δ 0)

Journal of Nanotechnology 3

interface of the three phases and the direction of theequilibrium shift of the TPL without the influence of thelinear tension

Using the trigonometric identity cos2φinfin0 + sin2φinfin0 1and excluding the angle ψ from equation (6) we express thegrowth angle φinfin0 of the NWs in an explicit manner asfollows

φinfin0 arccosα2S + α2L minus α2SL

2αSαL

1113888 1113889 (7)

According to (7) the angle φinfin0 is a constant value for thepresent crystallized substance under the given thermody-namic conditions of steady growth of NWs this is deter-mined not from a geometric viewpoint but rather accordingto the physical nature of the processes occurring within theneighborhood of the TPL e growth angle of the whiskersφinfin0 should be distinguished from the wetting angle θ0 whichcharacterizes the local equilibrium toward the movement ofa liquid droplet over the surface of the solid body and is notdirectly related to crystallization An exception is the case ofa catalyst droplet position with thermodynamic equilibriumon the crystal surface (Figure 2(b))

It follows from (7) that if αSgt αL and αLgt αSL to ensurethe circularity of the force vectors corresponding to the freesurface energies the equilibrium under the droplet must beconvex toward the droplet In contrast if αSlt αL the frontmust be concave If the curved section of the front crys-tallization below the droplet is concave (the curvature centeris outside the crystal) there is a high saturation above it(Kelvinrsquos effect) us this section transforms into the singleface of the crystallization front (Figure 1(a))

e angle ψ is expressed by the second part of equation(6) Instead of the growth angle φinfin0 we substitute its valuefrom (7) Hence we obtain the following

ψ arccosαL

αSL

1 minusα2S + α2L minus α2SL

2αSαL

1113888 1113889

2

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)

From equation (8) it is clear that with an isotropicapproximation the inclination angle (ψ) of the curvedcrystallization front of the NWs near the TPL is determinedby the equilibrium conditions at the interface border

According to equation (8) at φinfin0 0 the inclinationangle is ψ 90deg Hence the droplet in this case represents ahemisphere on top of the NWs and the radius of thecurvature of the TPL is maximized [10] Growth of the NWsunder these conditions is impossible because with an in-crease in the wetting perimeter of the growth substrate theinterface surface energy continuously decreases (full wet-ting) e radius of the line of the three-phase contact of thedroplet on the substrate tends toward infinity In this casethe liquid-phase catalyst spreads over the surface of thesubstrate and crystallization can lead to epitaxial growth ofthe film when a solution of an intermediate metallic sub-stance is applied to the substrate as a thin layer [26]

If φinfin0 90deg the contact angle φ is unstable and anycontact between the droplet and crystal leads to an increasein the interfacial energy (full nonwetting) e radius ofcurvature of the three-phase contact line is zero (r 0) andaccording to (8) ψ arccos(αLαSL) Because αLgt αS theangle ψ is not defined and stable growth of the NWs isimpossible At φinfin0 90deg a phase transition is only possible atthe border with the vapor [10 25]

At ψ 0 equation (8) moves toward a well-knowncondition of catalyst droplet equilibriumφinfin0 arcsin(αSLαL) which characterizes the nonwettingmodel for the stable growth of whiskers when a flat face ofthe crystallization front adjoins the TPL [10 26]

us an analysis of the thermodynamic angle of crys-tallization reveals that the growth of the NWs is thermo-dynamically possible if the contact angle φ is within theinterval 0ltφlt 90deg (90deglt θlt 180deg) At φge 90deg the contactangle of the droplet with the crystal is unstable and anycontact between the droplet and crystal leads to an increasein the free surface energy hence such contact is impossibleAt φle 0 no stable contact angle of the catalyst droplet occurson top of the NWs and in this case full wetting is realized[10 26]

Table 1 presents the calculation results for the growthangle φinfin0 the inclination angle of the curved surface of thecrystallization front ψ based on equation (8) and the contactangle θ0 based on equation (4) at χr 0 for certain pa-rameters of the growth of cylindrical Si GaAs and InAsNWs with the participation of Au and Ga particles thecalculation of angle θ and the following free energy values

αS

V

αL

LS

θ0

(a)

αS

αL

αSL

V

LS

φ0

(b)

Figure 2 Scheme of the three phases in the conjugation solid liquid and vapor (a) at a thermodynamic position of nonequilibrium butwith mechanical equilibrium of the droplet on the crystal surface and (b) at a position of thermodynamic equilibrium of the catalyst dropleton the crystal surface


αAuL 0910 Jm2 αS 12 Jm2 and αSL 045 Jm2 [2 24] forSi αAuL 0910 Jm2 αGaL 0800 Jm2 αAuSL (ZB) 0866 Jm2αAuSL (WZ) 0980 Jm2 αGaSL (ZB) 0780 Jm2 αS(ZB)

1500 Jm2 and αS(WZ) 1300 Jm2 [3] for GaAs andαS(ZB) 1190 Jm2 αS(WZ) 0910 Jm2 αAuL 1000 Jm2and αGaSL (ZB) 0800 Jm2 for InAs [3]

In our model the angle φinfin0 determines the direction ofthe TPL offset and represents the NW growth angle φinfin0(Figure 1(b)) under the condition of a constant growthangle When the volume of the catalyst droplet is main-tained the NWs grow in a cylindrical shape with an isotropicside surface However at certain angles ψ on the curvedcrystallization front under a droplet in the TPL neighbor-hood an internal inclined (oblique) tight-packed AAprime facemay occur characterized by small Miller indices hkl thusthe sidewall of the NWs will be inclined at a certain angle tothis face [27]

e appearance of such an AAprime face is possible owing tothe dependence of the specific free surface energy on theorientation of the face as it is known that the singular facescorrespond to a sharp minimum and the angular derivativezαzψ reaches the greatest value (Figure 3) [28]

So that a peripheral low-index face emerging into theTPL could appear at a point on the curved area of thecrystallization front under the catalyst droplet on the top ofNWs it is necessary that the front sections under the dropletnear this point do not serve as a source of steps for the faceis requires the initial curved crystallization front of NWsand the TPL itself to be convex (the curvature center of thesurface is situated in the crystal) If the formed singular facehkl comes out on the TPL then the latter changes its roundshape and straight sections should appear on it[8 27 29 30]

It is necessary that the initial curved crystallization frontof the NWs and the TPL is convex (the center of curvature ofthe surface is in the crystal) If the single face hkl reachesthe TPL the TPL changes from its round shape and adoptsstraight sections that should appear

e outside surface of the NWs is formed as a result ofthe TPL movement Hence under the appropriate condi-tions the emergence of oblique singular faces on the sidesurface of the growing NWs is energetically advantageous(eg faces (111) A and (111) C for crystals GaAs of thezincblende (cubic) crystal structure and the lt111gt orien-tation of the growth axis [8 30]) e exit of the inclinedfacets 111 on the sidewall surface in the untwined Si Geand GaAs NWs leads to facet formation with the crystal-lographic orientation 1―12 [27] e structure of the 1―12

facets is an interchange of (111) faces for Si and Ge and (111)A is an interchange for GaAs which is bent at an angle ofsim195deg with respect to the growth axis of the NWs and hasldquocorrectingrdquo steps between them e formation of the sidefaceting of NWs is described in [8 16 25 27 29 30]

A necessary condition for the appearance of an obliquetight-packed face (hkl) on the sidewall surface of the NWs isthat the equilibrium wetting angle of the oblique facetaccording to Youngrsquos modulus should increase the actualwetting angle of the facet θ minus ψ

During movement of the catalyst droplet along the in-clined surface (hkl) its radius and wetting perimeter de-crease while angle between the tangent to droplet surfaceand face θ minus ψ in condition of volume constancy of liquidphase increases consecutively until it reaches value θ(hkl) ofYoungrsquos in point Aprime When condition θ θ(hkl) is reached atsome point Aprime the further droplet movement along the face(hkl) is difficult due to increase of free surface energy in thisprocess after which the droplet flows onto the faceen thecrystallization starts on the oblique face and the TPL movesat angle θ minus φprimeinfin0 forming an expanding area of isotropicsurface of NWs Growth angle φ0prime is less than the initial angleφinfin0 as supersaturation on face (hkl) increases

e condition for the constancy of the growth angle φinfin0 in this case is satisfied only at the stage of growth of thetransition section (ldquocorrectingrdquo steps) between the twosections of the crystal narrowing along the face e facetNWs can be controlled by changing the wetting angle θprime ofthe catalyst droplet at the top of the crystal

4 Line Tension TPL

For the wetting model of NW growth the full equilibriumconditions of a closed line of the three-phase contact VLS atpoint A under a droplet can be obtained by introducing the

Table 1 Inclination angle of the curved surface of the crystallization front growth angle of Si GaAs and InAs NWs in the presence of Auand Ga particles and contact angle of the catalyst droplet on the flat surface

NW type Cubic Si (CUB)CubicGaAs(ZB)

Hexagonal GaAs (WZ) Cubic InAs (ZB) Hexagonal InAs (WZ)

Type of particles Au Au Ga Au Au GaGrowth angle φinfin0 in degrees 19 28 20 57 52 49Angle ψ in degrees 49 61 72 39 36 19Contact angle of droplet θ0 in degrees 34 46 37 69 84 82

ndashψ ψ0

αSL(hkl)

αSL

Figure 3 Anisotropy of crystalrsquos free surface energy nearby thesingular faces with small crystallographic indexes hkl


linear energy χr into the second equation of the system ofequations in equation (6) [10]

minus αS + αL cosφ0 + αSL sinψ 0

minus αL sinφ0 + αSL cosψ +χr

0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

At an angle φ0 in the system of equation (9) the truegrowth angle φ0 of the NWs differs from the angle φinfin0 inequation (6) for the case of a macroscopic crystal and dropletor a direct contact line ie for r⟶infin

By excluding the angle ψ from the system of equation (9)and considering equation (7) we express χr as follows

χr

αL sinφ0 minus

sin2φ0 minus2αS cosφinfin0 minus cosφ0( 1113857

αL

1113971

⎛⎝ ⎞⎠

(10)

Equation (10) provides the implicit dimensional de-pendence of the growth angle of NWs which is affected bythe linear tension during the heterogeneous formationprocess of a new phase According to equation (10) atφ0⟶φinfin0 and a constant αL the linear energy tends towardzero (χ⟶ 0) Additionally at the positive curvature of theTPL (where the center of the curvature is inside the dropletor crystal) the linear tension χ can be positive or negativee value of χ is positive ie its force contribution αχ atpointA on the TPL is directed radially inward when φinfin0 ltφ0A positive linear tension (χ gt 0) leads to a contraction of thecircular wetting perimeter of the droplet under a decrease inradius and an increase in angle φ0 compared with itsmacroscopic value φinfin0 Otherwise the linear tension isnegative (χ lt 0 and the force vector at point A on the TPL isdirected radially outward) and with a decrease in r a de-crease in φ0 occurs compared with φinfin0 which increases thecircular wetting perimeter

If the linear tension is positive a slight random change inthe NW diameter for example a decrease leads to an in-crease in the χr value hence a disbalance and decrease inthe droplet wetting perimeter occuren a reduction of theNW diameter is needed to restore the balance of the forceswhich leads to internally contracting growth and the for-mation of conical and needle crystals resulting in the dropletjumping off the top of the NW (Figure 4) Conversely arandom increase in the NW diameter leads to a decrease inthe χr value and further widening of the NWe describedmechanism of positive feedback causes unstable growth ofthe NWs

If χ lt 0 during NW growth a mechanism of negativefeedback is realized which stabilizes the crystal growth [7]In this case a slight disturbance reducing the radius of theNWs will reduce χr and lead to the recovery of the cross-sectional area of the crystal to the initial undisturbed stateHence the linear tension χ should be negative for stable NWgrowth

Using equation (10) we plot the dependence of thegrowth angle φ0 on χr (Figure 5) To construct the graph inFigure 5 the following values of the free surface energies were

used αL 091 Jm2 and αS 120 Jm2 which are typical forAu-Si systems [2 26]

As shown in Figure 5 the true growth angle φ0 of thewhisker increased in direct proportion to the dimensionlesscurvature χlowast However depending on the linear tension thethermodynamic growth angle φ0 could both increase (atχ gt 0) and decrease (at χ lt 0)

If we exclude the angle ψ from the system of equations(equation (9)and consider equation (7) we can express thethermodynamic growth angle of the NWs φ0 for a smallcatalyst droplet as follows

φ0 arccos cosφinfin0 +(χr)2 minus 2(χr)αL sinφ0

2αSαL

1113890 1113891 (11)

In deriving (11) on the basis of a detailed analysis of theradicals in a quadratic equation a negative sign was chosenbecause a positive sign yields nonphysical results It followsfrom (11) that at a positive curvature of the TPL and adecrease in the NW radius the growth angle φ0 can bothincrease and decrease in relation to the angle φinfin0

If χ gt 0 and (χr)gt 2 sinφ0 ie the force contribution ofthe linear tension in the droplet equilibrium is greater thanthe double vector projection corresponding to the freesurface energy of the liquidndashvapor interface on the plane ofthe droplet contour with a decrease in the crystal radius φ0decreases (the second term of the function argument of thearc cos on the right-hand side of (11) with a ldquo+rdquo sign)Conversely at χ gt 0 and χrlt 2 sinφ the growth angle φ0increases with a decrease in r (before the ldquominus rdquo sign appears inthe second term in (11)) For χ lt 0 as determined by theinfluence of the linear tension on the value the growth angledevelops in the opposite direction

For example for a dimensionless curvature of the TPLχlowast 02 and the aforementioned values αAuL and αS for Si atφinfin0 10deg the growth angle satisfying equation (11) is φ0 7degwhich is smaller than the given value of φinfin0 Hence in theconsidered example at a linear tension of χ lt 0 the TPL isstretched and the NWs should widen during growth For theaforementioned conditions but at χlowast 04 we obtain φ017degwhich is larger than φinfin0 10deg Hence it follows that χ gt 0and the NWs narrow during their growth

If the linear tension on the line of the interfacial contactis negative (χ lt 0) the equilibrium of the two-dimensionalphases for example those having free surface energies αSLand αLsinφ and that divided by a circular contour of thecatalyst droplet with a curvature radius r is stable isconclusion follows from the consideration of the extremumof the Helmholtz free energy with a change in the radius ofthe contour of the catalyst droplet on top of the NWs in atwo-dimensional system (in the action plane) for dr

dF 2πr αSL minus αL sinφ( 1113857 + 2πχ1113858 1113859dr 0 (12)

From equation (12) we obtain the following

αL sinφ minus αSL χr (13)

which is analogous to (5) at δlowast 0 en z2Fzr2

minus (2πχr)gt 0 as χ lt 0


A consideration of the linear tension does not affect theconclusion regarding the stability of NW growth if χ gt 0 asin the case of z2Fzr2 lt 0 [31]

5 Model Periodic Instability of NW Growth

e size dependence of the growth angle can explain theemergence of negative feedback in the system which leads tothe radial periodic instability of Si NWs (Figure 6) [32 33]Radial periodic instability is the result of the formation ofrhythmically repeated spikes and cavities on the sidewall ofthe NWs Periodic instability occurs only for the thinnestNWs (average diameter of 50ndash100 nm) Radial instabilitywas discovered by Givargizov in Si NWs grown using Auparticles [32] ese crystals consisted of a series of bundlesand overstretching with approximately equal intervals oc-curred between theme change in the crystal diameter wasdetermined by the instability of the wetting contour of the

droplet during growth Although the radius of the contourchanged periodically the bundles differed slightly from eachother with regard to diameter and length us the peri-odicity had a nonstrict nature and the sequences of bundlesof different diameters were not identical for differentcrystals Each crystal had its own law of oscillation Addi-tionally small amounts of AsCl3 vapor reduced the contactangle of the droplet and eliminated the instability

To describe the periodic instability of NW growth weanalyze the initial segment of a narrowing crystal At thenarrowing section toward the thick NWs (where the forcecontribution χr to the equilibrium at a point on the TPL canstill be small) the doubled projection of vector αL on theplane of the droplet contour (2αL sinφ0) exceeds the forcevector αχ aligned with the linear tension χ According toequation (11) the growth angle φ0 increases with a decreasein r An increase in φ0 leads to further narrowing of thecrystal a decrease in the wetting perimeter of the dropletand an increase in the vector projection corresponding tothe free surface energy αL (increase of 2αL sinφ0 in equation(11)) However as the wetting perimeter decreases (andhence the NW diameter decreases) the force contribution ofthe linear tension αχ grows faster than the projection of thevector αL onto the plane of the droplet contour is resultsin a decrease in the growth angle e decrease in φ0contributes to a shift of the TPL in the direction of the crystalextension erefore when a certain minimum diameter(diameter of overstretching) is reached the NWs againextend until reaching a maximum diameter (diameter of thebundle) e dynamic equilibrium becomes unstable andafter stopping the process repeats

6 Conclusions

It was shown that in the wetting growth regime of NWs ashift in the TPL occurs at a fixed angle with respect to thesurface of the catalyst droplet (angle of growth φ0) Addi-tionally the direction of the shift in the TPL is determined bythe zero increment of the free energy of the system in theaforementioned process because it is the direction of theequilibrium shift

(a) (b)

Figure 4 SEM image of needle NWs (a) Si and (b) GaAs

ndash10 ndash08 ndash06 ndash04 ndash02 0

10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ

Figure 5 Calculated dependences of the thermodynamic growthangle of Si NWs on the dimensionless curvature of the TPL χlowast

χαLr for different values of the whisker growth angle φinfin0 whichcharacterizes the macroscopic droplet (r⟶infin)


e dimensional dependence of the thermodynamicangle of NW growth is determined by the influence of thelinear tension on the process of the heterogeneousformation of a small crystalline phase It was shown thatat a positive curvature of the TPL with a decrease in theradius of the NWs the growth angle can either increaseor decrease in relation to the thermodynamic growthangle for a macroscopic catalyst droplet or a straightTPL

It was shown that the observed radial periodic insta-bility of Si NWs can be explained according to the di-mensional dependence of the thermodynamic growthangle which causes a negative feedback mechanism in thesystem Tip or needle NWs can be formed under the in-fluence of linear tension and the action of a positivefeedback mechanism

Data Availability

Any data and information used to support the findings ofthis study will be provided by the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e study was conducted with financial support from theRussian Foundation for Basic Research (RFBR) as part of thescientific project No 19-33-90219 ldquoNanoelectronics andNanotechnology Devicesrdquo at the collective center for use ofscientific equipment of Voronezh State Technical Universitye authors would like to thank Editage (httpwwweditagecom) for English language editing

References

[1] V Schmidt J V Wittemann and U Gosele ldquoGrowththermodynamics and electrical properties of silicon nano-wiresrdquo Chemical Reviews vol 110 no 1 pp 361ndash388 2010

[2] E I Givargizov ldquoFundamental aspects of VLS growthrdquoJournal of Crystal Growth vol 31 pp 20ndash30 1975

[3] V G Dubrovskii Nucleation 3eory and Growth of Nano-structures Springer-Verlag Berlin Germany 2014

[4] J Johansson L S Karlsson C P T Svensson et alldquoStructural properties of 〈111〉B-oriented III-V nano-wiresrdquo Nature Materials vol 5 no 7 pp 574ndash580 2006

[5] V A Nebolrsquosin J Johansson D B Suyatin andB A Spiridonov ldquoermodynamics of oxidation and re-duction during the growth of metal catalyzed silicon nano-wiresrdquo Journal of Crystal Growth vol 505 pp 52ndash58 2019

[6] V A Nebolrsquosin and A A Shchetinin ldquoA mechanism of quasi-one-dimensional vapor phase growth of Si and GaPwhiskersrdquoInorganic Materials vol 44 no 10 pp 1033ndash1040 2008

[7] V Schmidt S Senz and U Gosele ldquoe shape of epitaxiallygrown silicon nanowires and the influence of line tensionrdquoApplied Physics A vol 80 no 3 pp 445ndash450 2005

[8] D Jacobsson F Panciera J Tersoff et al ldquoInterface dynamicsand crystal phase switching in GaAs nanowiresrdquo Naturevol 531 no 7594 pp 317ndash322 2016

[9] B A Wacaser K A Dick J Johansson M T BorgstromK Deppert and L Samuelson ldquoPreferential interface nu-cleation an expansion of the VLS growth mechanism fornanowiresrdquo Advanced Materials vol 21 no 2 pp 153ndash1652009

[10] V A Nebolrsquosin A I Dunaev A F Tatarenkov andS S Shmakova ldquoScenarios of stable vapour-liquid droplet-solid nanowire growthrdquo Journal of Crystal Growth vol 450pp 207ndash214 2016

[11] V G Dubrovskii G E Cirlin N V Sibirev F JabeenJ C Harmand and PWerner ldquoNewmode of vaporminus liquidminus solidnanowire growthrdquo Nano Letters vol 11 no 3 pp 1247ndash12532011

[12] A Scheludko V Chakarov and B Toshev ldquoWater con-densation on hexadecane and linear tensionrdquo Journal ofColloid and Interface Science vol 82 no 1 pp 83ndash92 1981

(a) (b)

Figure 6 Radial periodic instability of Si NWs (a) [32] and (b) [34]


[13] V A Nebolrsquosin D B Suyatin A I Dunaev andA F Tatarenkov ldquoCapillary stability of VLS crystallizationprocesses and its comparison to Czochralski and Stepanovgrowth methodsrdquo Journal of Crystal Growth vol 463pp 46ndash53 2017

[14] V A Nebolrsquosin A I Dunaev and M A Zavalishin ldquoEffect ofthe line tension at the vaporndashliquidndashsolid boundary on thegrowth of silicon nanocrystalsrdquo Inorganic Materials vol 44no 6 pp 559ndash562 2008

[15] V A Nebolrsquosin D B Suyatin A I Dunaev S S ShmakovaM A Zavalishin and E V Ivannikova ldquoContribution of thefree energy of the three phase line of contact to the ther-modynamic equilibrium conditions of a metal solvent dropletin Si and Ge whisker growthrdquo Inorganic Materials vol 51no 3 pp 191ndash196 2015

[16] V A Nebolrsquosin A A Shchetinin A N Korneeva et alldquoDevelopment of lateral faces during vapor-liquid-solidgrowth of silicon whiskersrdquo Inorganic Materials vol 42 no 4pp 339ndash345 2006

[17] S Kodambaka J Tersoff M C Reuter and F M RossldquoDiameter-independent kinetics in the vapor-liquid-solidgrowth of Si nanowiresrdquo Physical Review Letters vol 96 no 9p 096105 2006

[18] S Kodambaka J Tersoff M C Reuter and F M RossldquoGermanium nanowire growth below the eutectic tempera-turerdquo Science vol 316 no 5825 pp 729ndash732 2007

[19] J S Rowlinson and B Widom Molecular 3eory of Capil-larity Dover Publications Mineola NY USA 2002

[20] B A Pethica ldquoContact anglesrdquo in Reports on the Progress ofApplied Chemistry vol 46 pp 14ndash17 Society of ChemicalIndustry London UK 1961

[21] B M Law S P McBride J Y Wang et al ldquoLine tension andits influence on droplets and particles at surfacesrdquo Progress inSurface Science vol 92 no 1 pp 1ndash39 2017

[22] V A Nebolrsquosin A A Shchetinin and E I NatarovaldquoVariation in silicon whisker radius during unsteady-stategrowthrdquo Inorganic Materials vol 34 no 2 pp 87ndash89 1998

[23] V V Voronkov ldquoOn the thermodynamic equilibrium at alinear junction of three phasesrdquo Fiz Tverd Tela vol 5 no 2pp 571ndash574 1963

[24] V A Nebolrsquosin A A Shchetinin T I Sushko andP Yu Boldyrev ldquoConstant growth angle for silicon whiskersrdquoInorganic Materials vol 32 no 9 pp 1031ndash1034 1996

[25] F Panciera Z Baraissov G Patriarche et al ldquoPhase selectionin self-catalyzed GaAs naniwiresrdquo Nano Lett 2020

[26] V A Nebolrsquosin D B Suyatin E V Zotova andS S Shmakova ldquoMetal solvent droplet stability in siliconwhisker growthrdquo Inorganic Materials vol 48 no 9pp 861ndash866 2012

[27] F M Ross J Tersoff and M C Reuter ldquoSawtooth faceting insilicon nanowiresrdquo Physical Review Letters vol 95 no 14p 146104 2005

[28] W K Burton N Cabrera and F C Frank ldquoe growth ofcrystals and the equilibrium of their surfacesrdquo PhilosophicalTransactions of the Royal Society A vol 243 pp 299ndash3581951

[29] V G Dubrovskii ldquoDevelopment of growth theory for vapor-liquid-solid nanowires contact angle truncated facets andcrystal phaserdquo Crystal Growth amp Design vol 17 no 5pp 2544ndash2548 2017

[30] V G Dubrovskii N V Sibirev N N Halder and D RitterldquoClassification of the morphologies and related crystal phasesof III-V nanowires based on the surface energy analysisrdquo 3e

Journal of Physical Chemistry C vol 123 no 30 pp 18693ndash18701 2019

[31] V A Nebolrsquosin A I Dunaev A S Samofalova andV V Korneeva ldquoContact interaction in an MndashSiO2 (M

metal catalyst for nanowhisker growth) systemrdquo InorganicMaterials vol 54 no 6 pp 558ndash563 2018

[32] E I Givargizov ldquoPeriodic instability in whisker growthrdquoJournal of Crystal Growth vol 20 no 3 pp 217ndash226 1973

[33] E I Givargizov and N N Sheftalrsquo ldquoMorphology of siliconwhiskers grown by the VLS-techniquerdquo Journal of CrystalGrowth vol 9 pp 326ndash329 1971

[34] S Q Feng D P Yu H Z Zhang Z G Bai and Y Ding ldquoegrowth mechanism of silicon nanowires and their quantumconfinement effectrdquo Journal of Crystal Growth vol 209 no 2-3 pp 513ndash517 2000


wets the facet of the crystallization front and does not wet asidewall surface of the NWs (nonwetting condition (ormode) of NW growth [10 11 13]) For a nonwetting growthmodel the linear energy that occurs when the catalystdroplet does not wet the sidewall surface of the NWs and aflat crystallization front appears under the droplet wascarefully analyzed in [7 14 15] However it is well knownthat at large contact angles of catalyst droplets on the top ofthe NWs at the crystallization front curved sections andeven oblique densely packed faces can appear leading to theformation of facets on the sidewall of the crystal [3 16] Anonplanar geometry of a crystallization front was realized inthe wetting model of NW growth [10 11] Nevertheless inthe literature the linear energy of a three-phase borderunder wetting conditions has not been described

In the present article we explain that under the wettingconditions for steady NW growth (without considering thekinetic effects) a shift in the three-phase line (TPL) of theinterface boundaries occurs at a fixed angle with the surfaceof the catalyst droplet Additionally we reveal the size de-pendence of the thermodynamic growth angle of the NWsdetermined according to the influence of the linear tensionof the three-phase contact vaporndashliquid crystal We alsoshow that the growth angle of the NWs should be distin-guished from the wetting angle which characterizes the localequilibrium toward the movement of the liquid droplet overthe surface of a solid body and is not directly related tocrystallization Furthermore we demonstrate that the ob-served radial periodic instability of the NWs can beexplained on the basis of the size dependence of the ther-modynamic growth angle which induces a negative feed-back mechanism in the system Tip or needle NWs can beformed under the influence of linear tension and the actionof a positive feedback mechanism

2 Wetting Model of NW Growth

e cross-sectional dimensions of NWs are usually designedaccording to the linear dependence on the size of the catalystdroplet As a rule the ratio of the wire diameter to thedroplet diameter is sim(07 09) [13 17 18] We assume thatduring the growth of cylindrical NWs the border of thethree-phase contact takes the form of a circle is corre-sponds to the thermodynamically stable form of a TPLwhich is the shortest length for a given area of a crys-talndashliquid interface

Assume that a liquid catalyst droplet with a certainvolume is placed on top of the NWs with a composition inaccordance with the phase diagram of a metal-ndashsemiconductor (Figure 1) As the geometrical parametersof the droplet we choose the base radius r and angles be-tween the tangent and the meniscus of the droplet of theliquid passing through some point A along the TPL withhorizontal θ and vertical φ (the NWs are grown in thevertical direction) Suppose that in the wetting regime [10]under the droplet a stepped crystallization front with athickness of z is formed Over time the TPL shifts in the z-direction according to the angle δ between the vertical andthe tangent or the angle ψ between the horizontal and the

tangent to the surface of NWs growing at point A (where ψis the inclination angle of the curved surface of the crys-tallization front)

en the shift of l from pointA to pointAprime is determinedas follows

l z

cos δ (1)

A shift in the TPL on l is accompanied by a reduction ofthe crystalndashliquid interface area by 2πrl sin δ an increase inthe surface crystal area by 2πrl a decrease in the liquidsurface area by 2πrlcos(φ+ δ) and a decrease in the length ofthe TPL by 2πrl sin δ Hence the change in the surface (ΔFS)and the linear component of the Helmholtz free energy(ΔFl) which is associated with the indicated shift of point Acan be expressed as follows

ΔF ΔFs + ΔFl 2πrlαS minus 2πrlαL cos(φ + δ)

minus 2πrlαSL sin δ minus 2πlχ sin δ(2)

where αS αL and αSL represent the free surface energies ofthe vaporndashcrystal vaporndashliquid and crystalndashliquid inter-faces respectively and χ represents the tension of a closedline of the three-phase VLS contact

In equation (2) the first term represents the increment inthe free surface energy of the crystal due to an increase in itssurface area through a variation in the droplet to height dzthe second term represents a decrease in the free surfaceenergy of the liquidndashvapor interface the third term repre-sents a decrease in the free surface energy of the crys-talndashliquid interface and the fourth term represents adecrease in the extra linear energy due to a reduction in theTPL length

By substituting (1) into (2) then differentiating (2) withrespect to angle δ and equating the derivative to zero(d(ΔF)dδ|δδlowast 0) we determine the shift in the TPLcorresponding to theminimum increment in the free surfaceand linear energies as follows

αS sin δlowast

minus αL sinφ + αSL +χr (3)

In equation (3) χr represents the two-dimensionalpressure αχ occurring from the bend of the contact line(analogous to the capillary Laplace pressure arising from thecurvature of the interface surface) [12] and 1r representsthe curvature of the line e absolute value of the lineartension χ is estimated to be between 1times 10minus 11 and 1times 10minus 9 Jm[7 19]

Equation (3) indicates that the angles δlowast and φ alongwith the free surface energies αS αL and αSL and the linearenergy αχ determine the equilibrium condition of thedroplet at each point of the TPL on top of the whisker

Equation (3) implies an equilibrium condition of thedroplet of a liquid lying on a horizontal solid surface (atδlowast 90deg)

αS αL cos θ0 + αSL +χr (4)

We obtained modified Youngrsquos equation for the me-chanical equilibrium of a droplet of liquid on a flat surface















S

zl

LV

R

A

2r

Aprime

φ0

αSLαχψ

θδ αS

αL

φ

(a)

S

zl

LV

R

2r

A

AprimeαSL αχ

αS

αL

φ = φ0

(b)






2αSαL

1113888 1113889 (7)




ψ arccosαL

αSL


2αSαL

1113888 1113889

2

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)







αS

V

αL

LS

θ0

(a)

αS

αL

αSL

V

LS

φ0

(b)














4 Line Tension TPL






ndashψ ψ0

αSL(hkl)

αSL






0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)



χr

αL sinφ0 minus


αL

1113971

⎛⎝ ⎞⎠

(10)









2αSαL

1113890 1113891 (11)
















6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)









































S

zl

LV

R

A

2r

Aprime

φ0

αSLαχψ

θδ αS

αL

φ

(a)

S

zl

LV

R

2r

A

AprimeαSL αχ

αS

αL

φ = φ0

(b)






2αSαL

1113888 1113889 (7)




ψ arccosαL

αSL


2αSαL

1113888 1113889

2

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)







αS

V

αL

LS

θ0

(a)

αS

αL

αSL

V

LS

φ0

(b)














4 Line Tension TPL






ndashψ ψ0

αSL(hkl)

αSL






0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)



χr

αL sinφ0 minus


αL

1113971

⎛⎝ ⎞⎠

(10)









2αSαL

1113890 1113891 (11)
















6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)































2αSαL

1113888 1113889 (7)




ψ arccosαL

αSL


2αSαL

1113888 1113889

2

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)







αS

V

αL

LS

θ0

(a)

αS

αL

αSL

V

LS

φ0

(b)














4 Line Tension TPL






ndashψ ψ0

αSL(hkl)

αSL






0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)



χr

αL sinφ0 minus


αL

1113971

⎛⎝ ⎞⎠

(10)









2αSαL

1113890 1113891 (11)
















6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)







































4 Line Tension TPL






ndashψ ψ0

αSL(hkl)

αSL






0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)



χr

αL sinφ0 minus


αL

1113971

⎛⎝ ⎞⎠

(10)









2αSαL

1113890 1113891 (11)
















6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)































0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)



χr

αL sinφ0 minus


αL

1113971

⎛⎝ ⎞⎠

(10)









2αSαL

1113890 1113891 (11)
















6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)

































6 Conclusions


(a) (b)



10

04 06 08 10

20

30

40

50

60

70

80

90

02

φ0 = φ0infin

φ0infin = 60deg

φ0infin = 30deg

φ0infin = 10deg

χ gt 0

χlowast

χ lt 0

φ0 rpaπ






Data Availability




Acknowledgments


References













(a) (b)






























Data Availability




Acknowledgments


References













(a) (b)




























development of growth theory for vapor–liquid–solid

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