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)
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3. Phase rule and phase diagrams(contd.)
Two component system
Case of Ga-As system
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3. Phase rule and phase diagrams(contd.)
Ternary system Pseudo-binary system
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Phase RulePhase Rule
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Miscibility gap
3. Phase rule and phase diagrams(contd.)
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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)
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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)
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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)
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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
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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
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