Appendix a DERIVATION of THERMODYNAMIC EQUATIONS

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Appendix A

DERIVATION OF THERMODYNAMIC EQUATIONS

This appendix derives the thermodynamic equations that are used in the main manuscript of our paper entitled “A Multi-scale Analysis of the Crystallization of Amorphous Germa- nium Telluride Using Ab-Initio Simulations and Classical Crystallization Theory”. Specifi- cally speaking, Equation (7), (8), and (9) in the main manuscript are derived, which quanti- tatively describe the three key physical variables that jointly determine the crystallization of amorphous PCM, i.e. the Gibbs free energy difference (Section A.1), the interfacial energy density (Section A.2), and the elastic energy density (Section A.3), respectively. In this appendix, we focus on mathematical derivation only, in order to provide more details about the theory that is used in the paper. The physical interpretation of the equations and various physical quantities can be found in the main manuscript.

A.1 Derivation of Gibbs free energy difference

The internal energy U (eV) of a thermodynamic system

U = U (S,V,Ni) (A.1)

3 is a function of entropy S (eV/K), volume (A˚ ), and number of atom species i (i =

1, 2, ··· ,N) Ni. Taking the total differentiation of internal energy, ∂U ∂U ∂U dU = dS + dV + dN (A.2) ∂S ∂V ∂N i V,Ni S,Ni i i S,V     X   If we define temperature T (K), pressure P (Pa), and chemical potential of atom species i

µi (eV) as ∂U ∂U ∂U T ≡ , P ≡− , µ ≡ , (A.3) ∂S ∂V i ∂N  V,Ni  S,Ni  i S,V respectively, we have

dU = T dS − PdV + µidNi (A.4) i X 2

The enthalpy H (eV) of a thermodynamic system is defined as

H = U + PV (A.5)

For the NPT system, the total differentiation of enthalpy

dH = dU + PdV (A.6)

= T dS + µidNi i X = T dS gives dH (T ) dS (T )= (A.7) T

By taking integration, we have

T dH (T ) S (T ) = S (Tm−)+ (A.8) − T ZTm T cV = S (Tm−)+ dT − T ZTm 3 where c is the specific heat capacity at constant pressure (eV/KA˚ ), and Tm is the melting temperature (K). Therefore,

T c s (T )= s (Tm−)+ dT (A.9) − T ZTm 3 where s is the entropy density (eV/KA˚ ), which is written as S hereafter for consistency with the main context. For amorphous GeTe,

T ca Sa (T )= Sa (Tm−)+ dT (A.10) − T ZTm 3 where Sa is the amorphous GeTe entropy density (eV/KA˚ ) and ca is the amorphous GeTe 3 specific heat capacity at constant pressure (eV/KA˚ ). For crystalline GeTe,

T cc Sc (T )= Sc (Tm−)+ dT (A.11) − T ZTm 3

3 where Sc is the crystalline GeTe entropy density (eV/KA˚ ) and cc is the crystalline GeTe 3 specific heat capacity at constant pressure (eV/KA˚ ). Therefore,

− ∆Sac (T ) = Sa (T ) Sc (T ) (A.12) T ca − cc = Sa (Tm−) − Sc (Tm−)+ dT − T ZTm T ca − cc = Sl (Tm+) − Sc (Tm−)+ dT − T ZTm ∆H T c − c = m + a c dT T − T m ZTm 3 where Sl is the liquid GeTe entropy density (eV/KA˚ ) and ∆Hm is the latent heat of fusion 3 of GeTe (eV/A˚ ). The Gibbs free energy of a system is defined as

G = U + PV − TS (A.13)

For the NPT system in an unconstrained state (P =0),

G = U − TS (A.14)

Therefore,

∆Gac (T ) = Ga (T ) − Gc (T ) (A.15)

= [Ua (T ) − Uc (T )] − T [Sa (T ) − Sc (T )]

= ∆Uac (T ) − T ∆Sac (T ) T ∆Hm ca − cc = ∆Uac (T ) − T + dT T − T  m ZTm  which is Equation (7) in the main manuscript.

A.2 Derivation of the interfacial energy density

2 To obtain the interfacial energy density σ (T ) (eV/A˚ ), the NPT AIMD is used to simulate three systems with periodic boundary conditions applied at the supercell boundaries:

× × a 1. amorphous GeTe supercell with size Lx Ly Lz . 4

× × c 2. crystalline GeTe supercell with size Lx Ly Lz.

× × a c 3. welded supercell with size Lx Ly (Lz + Lz) containing both amorphous and crys- talline GeTe supercells.

a c During the NPT simulation, Lx, Ly, Lz , and Lz are allowed to change to control pressure

P = 0. The internal energies of the three systems are Ua (eV), Uc (eV), and Uac (eV), respectively. For the NPT (P=0) systems, the Gibbs free energy (eV)

G = U + PV − TS (A.16)

= U − TS

3 where U is the internal energy (eV), P is the pressure (Pa), V is the volume (A˚ ), T is the temperature (K), and S is the entropy (eV/K). Since the NPT AIMD only last for several ps and no phase change will occur and entropy describes how disordered the system is, the welded system entropy (Sac) will be approximately the same as the sum of crystalline system entropy (Sc) and amorphous system entropy (Sa). Thus,

− − − Gac − (Ga + Gc) = Uac (Ua + Uc) T [Sac (Sa + Sc)] (A.17)

≈ Uac − (Ua + Uc)

Therefore, U − (U + U ) σ (T )= ac a c (A.18) 2A 2 i.e. Equation (8) in the main manuscript, where A is the interface area (A˚ ).

A.3 Derivation of the Murnaghan equation of state

The bulk modulus is defined as ∂P B = −V (A.19) ∂V  T and the pressure derivative of bulk modulus is

′ ∂B B = (A.20) ∂P  T 5

3 where V is the volume (A˚ ), T is the temperature (K), and P is the pressure (Pa). ′ ′ ′ Since B is experimentally found to change little with pressure, B = B0 (the subscript 0 means P = 0, i.e. the system has the most energetically favorable configuration). Thus, P ′ B = B0 + B dP (A.21) 0 Z ′ = B0 + B0P ∂P = −V ∂V  T Therefore, B0 −d P + ′ dV −dP B0 = ′ = (A.22) V B0 + B P ′  B0  0 B0 P + ′ B0 Taking the integration,   P V 1 B0 ln = − ln P + ′ (A.23) V0 B0 B   0 0 − 1 B0 B0 P + B′ = ln 0  −1 B0 B0 ′ B0 − 1  ′ B B0 0 = ln 1+ P B0   So, − 1 ′ B B0 0 V (P )= V0 1+ P (A.24) B0   i.e., B′ B0 V0 0 P (V )= − 1 (A.25) B′ V 0 "  # According to Equation A.4

dU = T dS − PdV + µidNi (A.26) i X the internal energy of crystalline GeTe is changed by

dU ≈−PdV (A.27) when the volume is change by dV . Here, we take advantage of that the entropy S is a description of how disordered the system is, thus −PdV is much more important than T dS and it dominates dU. 6

By taking the integration,

V U (V ) = U (V0) − PdV (A.28) V0 Z ′ V B V B0 V0 0 B0 = U (V0) − ′ dV + ′ dV V0 B0 V V0 B0 Z ′  Z B0 B0V ′ 1− ′ B0 0 1−B0 B0 = U (V0) − ′ ′ V − V0 + ′ (V − V0) B0 (1 − B0) B0  B′  B0V 1 V0 0 B0V0 = U (V0)+ + 1 − B′ B′ − 1 V B′ − 1 0 " 0   # 0 3 So, the elastic energy density (eV/A˚ ) is

U (V ) − U (V0) E (T ) = (A.29) e V B′ B0 1 V0 0 B0 V0 = + 1 − B′ B′ − 1 V B′ − 1 V 0 " 0   # 0 ′ ′ i.e. Equation (9) in the main manuscript. Since V0 = V0 (T ), B0 = B0 (T ), and B0 = B0 (T ),

Ee is temperature dependent too.