Modeling Temperature

Shyue Ping Ong

What is temperature?

Temperature is a measure of “excess” energy above the ground state due to excitations.

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Vibrational

Configurational

Electronic

Conformational

Approximating temperature effects

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ϕ = E +PV −TS −µO2NO2

Negligible for solids

Small compared to O2 entropy

Ong, S. P.; Wang, L.; Kang, B.; Ceder, G. Li−Fe−P−O 2 Phase Diagram from First Principles Calculations, Chem. Mater., 2008, 20, 1798–1807, doi:10.1021/cm702327g.

Temperature changes oxygen chemical potential

Vibrational entropy - Phonons

Collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids.

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Lattice dynamics of monoatomic 1D lattice

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(n-2)a (n-1)a na (n+1)a (n+2)a

un un+1 un+2 un-1 un-2

E0 ({un}) = E0 (0)+∂E∂un

"

#$

%

&'0

unn∑ +

12

∂2E∂un∂un)

"

#$$

%

&''0

unun)n, )n∑ +

13!

∂3E∂un∂un)∂un))

"

#$$

%

&''0

ununun))n, )n , ))n∑ +…

At equilibrium

E0 ({un}) ≈ E0 (0)+ 12

Dn, "n unun"n, "n∑ where Dn, "n =

∂2E∂un∂un"

%

&''

(

)**

0

Harmonic approximation M!!un = − Dn, "n un"

"n∑

un = Aei(qna−ωt )

Classical picture

Direct Approach – “Frozen” phonons

Explicitly calculate the forces between every atom and construct the force constant matrix of the crystal, and hence calculate normal modes of at any particular wavevector, q. Forces can be obtained in DFT using Hellman-Feynman Theorem Pros:

•  No specialized code required (except for automating displacements, etc.) •  Faster than linear response method, especially for reasonably sized systems. •  Many existing codes to help automate such computations: Phonopy, GoBaby, etc. •  Higher order anharmonic terms can be obtained relatively easily

Cons:

•  Large supercells are needed to accurately calculate the force constant matrix.

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∂E∂λ

= ψλ* ∂H∂λ

ψλ dV∫

Linear Response Method – Density Functional Perturbation Theory

From the Hellman-Feynman Theorem, we have

Linearizing the electron density, we get

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∂E∂λi

=∂Vλ (r)∂λi

nλ (r)dr∫

∂2E∂λi∂λ j

=∂2Vλ (r)∂λi∂λ j

nλ(r)dr+∫ ∂Vλ (r)

∂λi

∂nλ(r)

∂λ j

dr∫

Δn(r) = 4Re ψ*n (r)Δψn (r)

n=1

N /2

∑Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys., 2001, 73, 515–562, doi:10.1103/RevModPhys.73.515.

Linear Response Method – Density Functional Perturbation Theory

From first order perturbation theory, we have where

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(HSCF −εn ) Δψn = −(ΔVSCF −Δεn ) Δψn

ΔVSCF (r) = ΔV (r)+ e2 Δn( "r )

r− "rd "r∫ +

dvxcdn n=n(r)

Δn(r)

Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys., 2001, 73, 515–562, doi:10.1103/RevModPhys.73.515.

DFPT

Pros: •  Can calculate phonon frequencies at arbitrary wave vectors q

without use of supercells! •  Scaling with range of interatomic force constants is much more

favorable.

Cons:

•  Requires specialized codes •  Cost of calculations typically higher than frozen phonons

approach.

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Phonon Dispersions

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Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys., 2001, 73, 515–562, doi:10.1103/RevModPhys.73.515. Gonze, X.; Rignanese, G.-M.; Caracas, R. First-principle studies of the lattice dynamics of crystals, and related properties, Zeitschrift für Krist., 2005, 220, 458–472, doi:10.1524/zkri.220.5.458.65077.

Lattice dynamical properties

NANO266 11 Togo, A.; Chaput, L.; Tanaka, I.; Hug, G. First-principles phonon calculations of thermal expansion in Ti 3SiC2, Ti3AlC2, and Ti 3GeC2, Phys. Rev. B - Condens. Matter Mater. Phys., 2010, 81, 1–6, doi:10.1103/PhysRevB.81.174301.

Electronic entropy

Probablity is given by Fermi-Dirac function

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hiKS = −

12∇2 −

Zk

rik+

k∑ 1

2ρ(r')ri − r'

dr∫∫ '+Vxc[ρ(r)]

hiKSψi (r) = εiψi (r) Independent one-electron eigenstates can

be occupied or not

fi =e−β (εi−εF )

1+ e−β (εi−εF )

Sel = −kB fi ln( fi )+ (1− fi )ln(1− fi )[ ]i∑

Electronic configuration entropy and the phase diagram of LiFePO4

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Zhou, F.; Maxisch, T.; Ceder, G. Configurational Electronic Entropy and the Phase Diagram of Mixed-Valence Oxides: The Case of Li_xFePO_4, Phys. Rev. Lett., 2006, 97, 155704, doi:10.1103/PhysRevLett.97.155704.

Configuration Entropy – Ising Model

If J >0 -> Ferromagnetic ground state

If J<0 -> Anti-ferromagnetic ground state

In absence of magnetic field, system is permanently magnetized at low temperatures.

At Curie temperature, Tc, phase transition occurs between magnetic and paramagnetic phases (magnetization is zero).

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H (σ ) = − Jijσ iσ j<ij>∑ −µ hjσ j

j∑

Cluster expansion formalism

Generalization of Ising model

Partition function

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E σ( ) = V0 + Viσ ii∑ + Vijσ iσ j

i, j∑ + Vijkσ iσ j

i, j,k∑ σ k +…

Z = E σ( )σ

∑

Thermodynamic averages from Monte Carlo simulations

Sample states of a system stochastically with probabilities that match those expected physically

To perform the integral numerically,

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A = Aσe−βE (σ )

Zσ

∫ dσ

A = Aσe−βE (σ )

Zσ

∑ = Aσ p(σ )σ

∑

Simple sampling (random choice of states) is inefficient because thermodynamic probabilities are very sharply peaked (exponential term)!

Simple sampling

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p(σ)

States here contributes ≈0 to integral

Nearly all the contributions to integral comes from here

Is there a better sampling method?

Detailed balance

At steady state, flux between two states must be equal, i.e.,

If the attempt distributions are symmetric, i.e., random selection,

So we set

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p(m)π (m→ n) = p(n)π (n→m)where π (m→ n) = a(m→ n)A(m→ n)π is the transition matrix and is given by the product of the attempt distribution aand the acceptance distribution A.

a(m→ n) = a(n→m)p(m)A(m→ n) = p(n)A(n→m)A(m→ n)A(n→m)

=p(n)p(m)

= e−β (En−Em )

A(m→ n) = e−β (En−Em ) if p(n)< p(m)1 if p(n)> p(m)

#$%

&%

Metropolis algorithm for cluster expansions

1.  Start in state {σ1, σ2, …, σn}.

2.  Choose a new set of spins by “flipping” randomly selected σi* = -σi

3.  Calculate ΔE = E({σ1,…, -σi, …, σn}) - E({σ1,…, σi, …, σn})

4.  If ΔE < 0, accept σi*. If ΔE > 0, accept σi* with probability e-βΔE.

5.  Go back to step 1.

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Example: Modeling a atomic orderings in an alloy

Automated Cluster Expansions

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van de Walle, A.; Ceder, G. Automating First-Principles Phase Diagram Calculations, J. Phase Equilibria, 2002, 23, 348–359, doi:10.1361/105497102770331596.

Application Example – Temperature-dependent Phase Diagram of P2 NaCoO2

NANO266 21 Hinuma, Y.; Meng, Y.; Ceder, G. Temperature-concentration phase diagram of

P2-NaxCoO2 from first-principles calculations. Phys. Rev. B 2008, 77, 1–16.

State of the art in Cluster Expansions

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Van de Walle, A. A complete representation of structure-property relationships in crystals., Nat. Mater., 2008, 7, 455–8, doi:10.1038/nmat2200.

Compressive sensing paradigm for determining ECIs

Configurational dependence of property tensors

Nelson, L. J.; Hart, G. L. W.; Zhou, F.; Ozoliņš, V. Compressive sensing as a paradigm for building physics models, Phys. Rev. B, 2013, 87, 035125, doi:10.1103/PhysRevB.87.035125.