First Principles Thermodynamics in Nanomaterials: Applications to Surfaces

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First Principles Thermodynamics in
Nanomaterials: Applications to Surfaces

L. Liborio

Computational Materials Science Group

DFT Review

Write the electronic density in terms of a set of non
-
interacting orbitals:


kinetic energy


nuclei
potential


electrostatic
interaction.


exchange and
correlation

If E
xc
[

] were known, the
exact

ground state could
be found.


E
Tot
=T
s
+E
ee
+E
ne

+E
nn
+T
n

Thermodynamics Review

P
0
, T
0
,
V
0
, U
0

gas

P
f
, T
f
,
V
f
, U
f

gas

Q

W

Examples of processes:

a) dU=0 (Complete cycle)

b) dU=0 (W=
-
Q, steady state)

c) dU=W (Q=0, thermal insulation)

First principle:

Natural and Reversible
processes

Second principle:

Reversible process, closed phase, no
chemical reactions, absorbs Q and
performs W.

U is also known as a
Characteristic Thermodynamical
function
.

Thermodynamics Review

Helmholtz free energy: F=U
-
TS, independent variables (T,V)

Enthalpy: H=U+PV, independent variables (S,P)

Gibbs Free Energy: G=U
-
TS+PV, independent variables (T,P)

If, for a given P and T, G(T,P) is a minimum, then the system is
said to be in a stable equilibrium.

DFT allow for the
calculation of the
total energy of a
nanosystem

This energy can
be linked to the
internal energy,
U, from
Thermodynamics

U can be used to
define the Gibbs
free energy, G,
of the
nanosystem

G can be used
to study the
stability of the
nanosystem

First Principles Thermodynamics

Nanosystems

Unit cell

Lattice param.

Surface

Defective bulk

Crystalline structures: atoms are
arranged in a periodic spatial
arrangement

Metals

Ceramics

Oxides

Atomic Scale surface reconstructions in a Ceramic: Strontium Titanate (SrTiO
3
).

Neutral oxygen defects in an Oxide: Titanium Dioxide (TiO
2
) in the rutile structure.

Strontium Titanate

Ti

O

Sr


Substrate for superconducting
thin films.


Buffer material for the growth of
Ga As on Si.

(1x1)
-
TiO
2

terminated surface

(1x1)
-
SrO terminated surface

(001)

Overview of the problem

Castell’s


model

Double
layer model

Sr
-
adatom


model

M. Castell in Surface
Science 505 (2002) 1
-
13

c(4x2) surface reconstruction

Overview of the problem


A great variety of surface reconstructions have been
observed, namely:
(2x1)
,
c(4x2)
[1][2][3],

(2x2)
,
c(4x4)
,
(4x4)
[1][2],

c(2x2)
,
(
√5x
√5)
,
(√13x√13)
[1].

And several structural models have been proposed, among
which are the ones presented in the previous slide.

Under which circumstances are any of these models
representing the observed surface reconstructions? Are any
of these in equilibrium?


[1]

T.Kubo and H.Nozoye, Surf. Sci. 542 (2003) 177
-
191.

[2] M.Castell, Surf. Sci. 505 (2002) 1
-
13.

[3] N. Erdman
et al,
J. Am. Chem. Soc. 125 (2003) 10050
-
10056.


Calculation Technique


Simulations within DFT theory using LDA approximation (T=0K)


Core electrons replaced by Troullier
-
Martin pseudopotentials


Calculations were carried out using the SIESTA program


Static calculations to predict equilibrium states (minimun energy)


Geometry:



Reconstructions using SrTiO
3
bulk
lattice constant



7
-
layer slabs separated by 3 layers of
vacuum



3 outermost layers fully relaxed

Thermodynamics of Surface Reconstructions

Surface excesses:


(1x1)TiO
2
-
terminated


O
= 0

(2x1)Ti
2
O
3
-
terminated


O
=
-
1/2

Components of the system:

SrO
, TiO
2
,O

O
2

SrO

TiO
2
O
2

O
2

Thermodynamics of Surface Reconstructions

Gibbs free energy definition:

Thermodynamics of Surface Reconstructions

Oxygen Gibbs free energy

We used 12 oxides: SrO,
TiO
2
, MgO, SiO
2
, Al
2
O
3
,
CaO, PbO
2
, CdO, SnO
2
,
Cu
2
O, Ag
2
O, ZnO

Experimental Value

Thermodynamics of Surface Reconstructions


First principles +
analytical expression

The dependence of the surface energy with p and T comes through the gas phase.

Calculated from first principles

Results: Kubo and Nozoye

T. Kubo and H. Nozoye, Surface Science 542 (2003) 177
-
191


(1x1)
Θ
=1


(2x1)
Θ
=0.5


c(4x2)
Θ
=0.25

Coverage
Θ

As we increase the
temperature,


tends
to decrease (not
monotonically) as the
surface goes through
a sequence of
reconstructions.

UHV=5x10
-
12

atm

~1200K

~1500K

Results: Kubo and Nozoye

0
: TiO
2
-
terminated (1

1)

=0
,

1
: (

13

13)

=0.0769,


2
: c(4

4)

=0.125,


3
: (

5

5)

=0.20,

4
: (2

2)

=0.25

.


L. Liborio,
et al
. J. Phys.: Condensed Matter 17. L223
-
L230. 2005

Results: Kubo and Nozoye

Equilibrium with SrO

0
: TiO
2
-
terminated (1

1)

=0
,

1
: (

13

13)

=0.0769,

2
: c(4

4)

=0.125,


3
: (

5

5)

=0.20,
4
: (2

2)

=0.25

.

4

2

3

1

~1200K

~1500K

Conclusions


We have calculated the surface energy of the Sr adatom structures. These
structures were proposed by Kubo and Nozoye to explain a set of structural
phase transitions on the SrTiO
3

(001) surface. The different surface structures
were observed using an STM.


Only the surface with coverage


=0.20 is stable for the ranges of
temperature and pressure reported by Kubo and Nozoye. Our calculations
show that the lower Sr coverages implied in the Sr adatom model can only be
explained if the surface is far from equilibrium, in a transient state as it loses
Sr to the enviroment.