Lattice Constants in Materials Using DFT Calculations

DFT CALCULATIONS FOR SIMPLE

DFT CALCULATIONS FOR SIMPLE

SOLIDS

SOLIDS

2.1 PERIODIC STRUCTURES, SUPERCELLS, AND

2.1 PERIODIC STRUCTURES, SUPERCELLS, AND

LATTICE PARAMETERS

LATTICE PARAMETERS

The vectors that define the cell volume and the

The vectors that define the cell volume and the

atom positions within the cell are collectively

atom positions within the cell are collectively

referred to as the 

referred to as the 

supercell

supercell

, and the definition of a

, and the definition of a

supercell is the most basic input into a DFT

supercell is the most basic input into a DFT

calculation.

calculation.

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How can we use calculations of this type to

How can we use calculations of this type to

determine the lattice constant of our simple cubic

determine the lattice constant of our simple cubic

metal that would be observed in nature?

metal that would be observed in nature?

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The simplest approach is to write the total energy using a

The simplest approach is to write the total energy using a

truncated Taylor expansion:

truncated Taylor expansion:

By definition, 

By definition, 

α

α

=0 if a

=0 if a

0

0

 is the lattice parameter

 is the lattice parameter

corresponding to the minimum energy.

corresponding to the minimum energy.

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This suggests that we can fit our numerical data to :

This suggests that we can fit our numerical data to :

The solid black curve shown in Fig. 2.1 is the result of fitting

The solid black curve shown in Fig. 2.1 is the result of fitting

this curve to our data using values of a from 2.25 to 2.6Å. This

this curve to our data using values of a from 2.25 to 2.6Å. This

fitted curve predicts that a0 is 2.43Å.

fitted curve predicts that a0 is 2.43Å.

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2.2 FACE-CENTERED CUBIC MATERIALS

2.2 FACE-CENTERED CUBIC MATERIALS

We define fcc metal cell vectors:

We define fcc metal cell vectors:

These vectors define the fcc lattice if we place atoms at

These vectors define the fcc lattice if we place atoms at

positions:

positions:

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8

9

F

F

r

r

o

o

m

m

t

t

h

h

e

e

c

c

u

u

r

r

v

v

e

e

i

i

n

n

F

F

i

i

g

g

.

.

2

2

.

.

3

3

w

w

e

e

p

p

r

r

e

e

d

d

i

i

c

c

t

t

t

t

h

h

a

a

t

t

a

a

0

0

=

=

3

3

.

.

6

6

4

4

Å

Å

,

,

e

e

x

x

p

p

e

e

r

r

i

i

m

m

e

e

n

n

t

t

a

a

l

l

l

l

y

y

,

,

t

t

h

h

e

e

C

C

u

u

l

l

a

a

t

t

t

t

i

i

c

c

e

e

c

c

o

o

n

n

s

s

t

t

a

a

n

n

t

t

i

i

s

s

3

3

.

.

6

6

2

2

Å

Å

.

.

2.3 HEXAGONAL CLOSE-PACKED MATERIALS

2.3 HEXAGONAL CLOSE-PACKED MATERIALS

The supercell for an hcp metal can be defined using the

The supercell for an hcp metal can be defined using the

following cell vectors:

following cell vectors:

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Using DFT to predict the lattice constant of Cu in the

Using DFT to predict the lattice constant of Cu in the

simple cubic or fcc crystal structures was

simple cubic or fcc crystal structures was

straightforward; we just did a series of calculations

straightforward; we just did a series of calculations

of the total energy as a function of the lattice

of the total energy as a function of the lattice

parameter, a. The fact that the hcp structure has 

parameter, a. The fact that the hcp structure has 

two

two

independent parameters, a and c

independent parameters, a and c

, complicates this

, complicates this

process.

process.

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One way to proceed is to simply fix the value of c/a and

One way to proceed is to simply fix the value of c/a and

then calculate a series of total energies as a function of a.

then calculate a series of total energies as a function of a.

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The energy functional can be written as 

The energy functional can be written as 

：

：

Where we have split the functional into a collection of

Where we have split the functional into a collection of

terms we can write down in a simple analytical form,

terms we can write down in a simple analytical form,

E

E

known

known

[{

[{

Ψ

i

i

}], and everything else, E

}], and everything else, E

XC

XC

. The“known” terms

. The“known” terms

include four contributions:

include four contributions:

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2.4 CRYSTAL STRUCTURE PREDICTION

2.4 CRYSTAL STRUCTURE PREDICTION

The main message from this discussion is that DFT is

The main message from this discussion is that DFT is

very well suited to predicting the energy of crystal

very well suited to predicting the energy of crystal

structures within a set of potential structures

structures within a set of potential structures

, but

, but

calculations alone are almost never sufficient to

calculations alone are almost never sufficient to

truly 

truly 

predict new structures 

predict new structures 

in the 

in the 

absence of

absence of

experimental data

experimental data

.

.

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2.5 PHASE TRANSFORMATIONS

2.5 PHASE TRANSFORMATIONS

The Gibbs free energy can be written as:

The Gibbs free energy can be written as:

where E

where E

coh

coh

, V, and S are the cohesive energy ,

, V, and S are the cohesive energy ,

volume, and entropy of a material.

volume, and entropy of a material.

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If we are comparing two possible crystal structures, then we

If we are comparing two possible crystal structures, then we

are interested in the change in Gibbs free energy between

are interested in the change in Gibbs free energy between

the two structures:

the two structures:

In solids, the first two terms tend to be much larger than

In solids, the first two terms tend to be much larger than

the entropic contribution from the last term in this

the entropic contribution from the last term in this

expression, so

expression, so

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