1Thermodynamics

Course: Advanced semiconductor materials, (S.Lourdudoss)

Source:1) R.F.Brebick, Phase Equilibria, ch. 2 in Handbook of Crystal Growth, vol. 1a, Fundamentals, Thermodynamics an Kinetics, ed. D.T.J.Hurle, North Holland, Elsevier, Amsterdam, 1993, ISBN 0-444-88908-62) G.B.Stringfellow, Organometallic vapor-phase epitaxy, 2nd edition, Academic press, NY, 1999.

Topics1. Thermodynamics vs. kinetics2. Basic thermodynamics of phase equilibrium3. Phase diagrams

21. Thermodynamics vs. kinetics Thermodynamics can tell the feasibility of a process, i.e., ideally what can be achieved under the specified conditions Kinetics can tell the reality of a process, i.e., steps involved, the rates of each step etc. It follows: knowledge on both thermodynamics and kinetics is necessary Basic goal of thermodynamics as applied to epitaxy: To define the compositions of the various phases in an equilibrium system at constant temperature and pressure. E.g., what is GaxIn1-xAsyP1-y if T, P and III/V ratio in the gas phase are givenTo assess the thermodynamic stability of the solid phases formed To assess the dopant incorporation

32. Basic thermodynamics of phase equilibrium

General:G = H TS where H = E + PV

At constant temp.: G = H - TS

When are two phases 1 and 2 (both containing e.g., two species A and B) in equilibrium?

If A in 1 = A in 2 andB in 1 = B in 2i is called the chemical potential of species i; it is a partial molar derivative of Gibbs free energy, G

jnPTi

i nG

,,

=

G (molar free energy) =

xAA+ xBB

4Chemical potentials of Chemical potentials of A and B in a mixture AA and B in a mixture A--B at B at xxBB

G (molar free energy) = xA A + xB B (1) Gibbs- Duhem equation:

xA dA + xB dB = 0 at const T and P (2) Differentiate (1) w.r.t. xB:

(dG/dxB)T, P = B - A (3)= Slope of the tangent at xB

N.B.: A and B are the chemical potentials for the composition at xB

5EQULIBRIUM CONDITION FOR TWO PHASES

Eq. of the tangent passing through xB' in G-x diagram (Gibbs isotherm):(y - G') = (B' - A' ) (xB - xB') (4)

G' from (1) in (4):y (xA' A' + xB' B') = (B' - A' ) (xB - xB')ory = (xA' A' + xB' B') + (B' - A' ) (xB - xB') (5)

At xB' = xB = 0: y-intercept is A'At xB' = xB = 1: y-intercept is B'

=> A COMMON TANGENT AT xB' (phase 1) and xB''(phase 2) MEANS SAME A AND B FOR BOTH PHASES = EQUILIBRIUM

62. Basic thermodynamics of phase equilibrium (contd.)

i in ideal gas mixturesi = i0 + RT ln (pi/pi0)pi = partial pressure of ipi0 = vapour pressure of pure component

i in ideal liquid or solid solutions i = i0 + RT ln (xi)xi = mole fraction of i

i in non-ideal liquid or solid solutions not well known i = i0 + RT ln (xii) = i0 + RT ln (ai) = activity coefficient, a = activity a or not knownRemedy: Develop a model (Several models described in Stringfellow)

72. Basic thermodynamics of phase equilibrium (contd.)

When does phase segregation arise? Same rules on slide 3 also explain phase segregation in an alloy

Case study of A-B formation:1) If EAB < ( EAA+EBB) where E = energy of interaction, then free energy of mixing very large and negative with respect to that of the ideal mixture=> phase segregation not favoured 2) If EAB = ( EAA+EBB) then free energy of mixing is still negative but not as large as case 1 above=> phase segregation not favoured3) If EAB > ( EAA+EBB), =>phase separation feasible

f = mole fraction

Solution

Solution

SolutionMechanical

mixture

Mechanical mixture

Mechanical mixture

8Binodal and spinodal decompositions

2. Basic thermodynamics of phase equilibrium (contd.)

A BC D

The locus of inflexion points C and D from G-x curve in the T-x curve = spinode

93. Phase rule and phase diagrams

Phase rule: F = C P + 2

F = no. degrees of freedomC = no. of componentsP = no. of phases2 = Temperature and Pressure

One component system Case of water (schematic)

10

3. Phase rule and phase diagrams(contd.)

Two component system

Case of Ga-As system

11

3. Phase rule and phase diagrams(contd.)

Ternary system Pseudo-binary system

12

Phase RulePhase Rule

13

Miscibility gap

3. Phase rule and phase diagrams(contd.)

14

3. Phase rule and phase diagrams(contd.)

Coherency strain (i.e., elastic strain without dislocations) stabilises the homogeneous solid!

(Lowers the Tc above which homogeneous solid exits)

15

Driving force for crystal growthDriving force for crystal growth

Consider the following reaction (in a closed chamber):M (s) + aA (g) = bB (g) + cC (g) + dD (g) (1)a, b, c, d = reaction coefficients

At equilibrium:At equilibrium:M,solid + aA = bB + cC + dD (2)M,solid = - aA0 + bB0 + cC0 + dD0 + RT ln (PA,eq-a PB,eqb PC,eqc PD,eqd)

(3)where Pi,eq = equilibrium pressureEqm. Constant of the reaction , K(T) = PA,eq-a PB,eqb PC,eqc PD,eqd (4)Therefore, (4) becomes

M,solid = - aA0 + bB0 + cC0 + dD0 + RT ln K(T) (5)

16

Driving force for crystal growth (contd)Driving force for crystal growth (contd)

At nonAt non--equilibrium:equilibrium:We assume a hypothetical chemical potential of the solid M, M,gas

M,gas + aA = bB + cC + dD (6)

M,gas = - aA0 + bB0 + cC0 + dD0 + RT ln (PA-a PBb PCc PDd) (7)where Pi = nonequilibrium pressure (e.g. Input pressure)

Growth: M,gas > M,solid (8)Etching: M,gas < M,solid (9)

Driving force for crystal growth: = M,gas - M,solid = RT ln[PA-a PBb PCc PDd/K(T)] (10)

17

3. Phase rule and phase diagrams(contd.)

MOVPE phase diagrams

=> Solid: x = 0.8, y = 0.59Gas: Ga/(Ga+In) = 0.8, As/(As+P) = 0.7

18

3. Phase rule and phase diagrams(contd.)

VPE phase diagram MBE phase diagram

Deposition diagram for GaxIn1-xAsyP1-y. The dashed line = solutions lattice matched to InP. From Koukitu and Seki, J. Crystal Growth, 49 (1980) p325.A=Cl/(inert gas+H2); B=(III-V)/(inert gas+H2); F=H2/inertgas; R3=GaCl/(GaCl+InCl); R5 =AsH3/(AsH3+PH3)

GaAs phase diagram at a total pressure of 10-5 torr