1

L2 Two-component systems (ii)

• The surface of a crystal is a cross discontinuity and has a free energy associated with it. • The value of this free energy depends on the orientation of the face and on the other phase in contact (vacuum, liquid).• A first approximation of the surface free energy

selected materials arecompiled in Table 2.2

2

L2 Two-component systems (iii)

• The equilibrium shape of a crystal consists of a set of singular faces by using the principle of surface potential minimization (Wulff construction)

3

L2 Two-component systems (iv)

4

L2 Nearly ideal solid solutions (i)

• Mixed crystals like Ge1-xSix, In1-xGaxAs , Al1-xGaxN ⇒ increasing importance

• Monocrystals with high compositional homogeneity (x uniformity) ⇒ required

• From the phase diagram: growth conditions- growth technique, T, P and concentration can be determined

• An ideal solution ⇒ a system with complete miscibility in both the liquid and solid phase for the whole compositional range 0<x<1.

• In the low pressure T-x projection ⇒ three fields: liquid, liquid + solid, and solid, separated by two boundaries ⇒ liquidus L and solidus S (fig. 2.6).

• The liquidus and solidus lines can be calculated theoretically.

5

L2 Nearly ideal solid solutions (ii)

• With respect to the component i=B, the equilibrium between solid and liquid phases is given by

6

L2 Nearly ideal solid solutions (iii)

• Fig. 2.6 shows the T-x projection of the Ge-Si phase diagram.

• Since the experimental and theoretical curves agree well ⇒ one can say that the system exhibits nearly ideal mixing behavior.

• This is also reflected in the near linear variation of the lattice constant with composition ⇒ Vegard’s rule.

• An important characteristic of the system can be determined from the phase diagram ⇒ Ko (thermodynamic) equilibrium distribution coefficient of component B.

XBs – concentration of B in solid, XBl – concentration of B in liquid, ∆hB

o – standard enthalpy and TmB melting temperature of the pure component B, R-constant.

7

L2 Systems with compound formation

• The tendency to form intermediate phases arises from the strong attractive forces between unlike atoms.• In such phases the bonds have much stronger ionic or covalent character compared to metallic ordered solutions ⇒ the free energies, i.e. entropy and enthalpy are typically low leading to their high stability.• The following chemical reaction takes place: mA(s) + nB(s) = AmBn(s)

In this case m=n=1

• Due to the relatively narrow stability region the compound is shown as a single vertical line of exact stoichiometry ⇒ small deviations determine a region called range of existence or range of homogeneity. • In this region the material is still stoichiometric and contains a certain number of intrinsic defects (equilibrium).

• Semiconductor compounds:Width of existence 10-5 -10-2 mole fractionFormation enthalpy of point defects 1- 4eV

8

Phase diagram Si-C

No liquid phasesublimation from 1800°Csolubility of carbon in liquid Si 0,026 at% at 1700°C

0 20 40 60 80 100Si CAt % C

Graphite+

SiC

V+graphiteV

L

V+L

L+SiC

V+SiC

SiC

Si+SiC

T°C

2830

1402

Growth is a high temperatureprocess

P = 1 atm

Issues: crucible material, heating, tightness, purity, etc.

L2

9

L2

10

L2

11

L2

• The melt growth processes are characterized by relatively low driving forces ⇒ ∆H/Tm usually falls within the range of 10 to 100Jmol-1; ∆T is typically 1-5K

⇒ ∆µ l→s is about 10-100Jmol-1.

12

L2

• Because in melt growth and LPE the crystals and layers are grown in near equilibrium conditions ⇒ the actual equilibrium states as described by the phase diagrams can be used as good approximation.• At high deviation from equilibrium ⇒the equilibrium phase diagrams are a rough tool only

13

L2

14

L2

15

L2

16

L2

• The nucleus can grow only if its radius exceeds the critical one ( due to a decrease of the total free energy• The nucleus is unstable at r<rcrit⇒ Under this condition it will dissociate• The critical nucleus radius ⇒rcrit = 2σΩ/∆µ

This is the case of homogeneousnucleation

• Pr – vapor pressure over a droplet with radius r• P∞ - equilibrium pressure over flat surface

Vm ≡ Ω – molar volume

17

L2

Two dimensional nucleation (i)

• In the practice of bulk and thin film crystal growth nucleation is initiated on a seed crystal or on a single crystalline substrate ⇒ the case of homogeneous three-dimensional nucleation in a metastable fluid phase is not relevant (important is for industrial mass crystallization).• 2D nucleation is the main growth mode for epitaxial layers ( homo- and heteroepitaxy), and for bulk crystal growth at atomically flat surfaces ⇒ after the 2D-nucleus is formed, the whole interface plane is then rapidly completed by lateral growth.• The change of the free Gibbs potential at the generation of a disc-shaped nucleus is expressed by the energy balance:

18

L2

Two dimensional nucleation (ii)

• The figure shows the critical radius of disc-shaped 2D-nucleus versus supersaturation for different methods of Si homoepitaxy at 500K using eq. (2.35).• In the case of highly supersaturated growth from the vapor (MOCVD, MBE) the nucleus dimension yields only a few nm, which is comparable to a cluster consisting of a few atoms only.• Crystal growth from the melt occurs mainly under conditions of an atomically rough interfaces, which are growing without nucleation generation, hence under very small supercooling ( i.e. driving force). In certain cases, parts of the interface can grow with atomically flat interface-forming facets ⇒ a discontinuous nucleation mechanism takes place ⇒ twins can be formed.• In epitaxial processes, where mainly atomically smooth interfaces occur, the growth mode of 2D nucleation is typically prevented by using off-oriented (vicinal) substrates ⇒ continuous step-flow mechanism