Transcript Slide 1

First Principles Thermodynamics in
Nanomaterials: Applications to Surfaces
L. Liborio
Computational Materials Science Group
DFT Review
ETot=Ts+Eee+Ene+Enn+Tn
Write the electronic density in terms of a set of non-interacting orbitals:
 (r )   i (r )
2
i
E elec[  ]  Ts [  ]  Eext[  ]  EH [  ]  Exc[  ]
kinetic energy
If Exc[] were known, the
exact ground state could
be found.
nuclei
potential
electrostatic
interaction.
exchange and
correlation
E xcLDA [  ]    ( r ) xc (  ( r )) dr
Thermodynamics Review
First principle:
W
W  Q  dU
Examples of processes:
gas
gas
P0, T0,
V0, U0
Q
a) dU=0 (Complete cycle)
b) dU=0 (W=-Q, steady state)
Pf, Tf,
Vf, Uf
Reversible process, closed phase, no
chemical reactions, absorbs Q and
performs W.
c) dU=W (Q=0, thermal insulation)
Natural and Reversible
processes
dU  TdS  PdV
Second principle:
U is also known as a
Characteristic Thermodynamical
function.
dS 
Q
T
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
First Principles Thermodynamics
G can be used
to study the
stability of the
nanosystem
Nanosystems
Crystalline structures: atoms are
arranged in a periodic spatial
arrangement
Metals
Ceramics
Oxides
Unit cell
Lattice param.
Defective bulk
Surface
Atomic Scale surface reconstructions in a Ceramic: Strontium Titanate (SrTiO3).
Neutral oxygen defects in an Oxide: Titanium Dioxide (TiO2) in the rutile structure.
Strontium Titanate
(001)
Sr
Ti
O
(1x1)-TiO2 terminated surface
•
•
Substrate for superconducting
thin films.
Buffer material for the growth of
Ga As on Si.
(1x1)-SrO terminated surface
Overview of the problem
M. Castell in Surface
Science 505 (2002) 1-13
Double
layer model
Castell’s
model
Sr-adatom
model
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 SrTiO3 bulk
lattice constant
• 7-layer slabs separated by 3 layers of
vacuum
• 3 outermost layers fully relaxed
Thermodynamics of Surface Reconstructions
O2
SrO TiO2 O2
O2



Gs   gi Ni 
i


(1x1)TiO2-terminated
 O= 0
(2x1)Ti2O3-terminated

1
 gTiO2 , pO2 , T 
2 As
Surface excesses:
1
i 
As

N ibulk 
 N i  N SrO N bulk 

SrO 
Components of the system:
SrO, TiO2,O
O= -1/2
Thermodynamics of Surface Reconstructions
  pO , T , gTiO  
2
2
1
2 As
1


bulk
G

N
g

g


g
(
T
,
p
)

SrO SrTiO 3
TiO2 TiO2
O
O2
O2 
 s
2 2

Gibbs free energy definition:
G  U  TS  pV
0K
( Es0 K  N SrTiO 3 ESrTiO
 TiO2 ETiO2 ) 
3
( Es (T )  N SrTiO 3 ESrTiO 3 (T )  TiO2 ETiO2 (T )) 
 T .(S s  N SrTiO 3 S SrTiO 3  TiO2 STiO2 ) 
 p.(Vs  N SrTiO 3VSrTiO 3  TiO2VTiO2 ) 
1
 O2 g O2 (T , p)
2
Thermodynamics of Surface Reconstructions
  pO , T , ETiO
2
2

1  0K
1

0K

(
E

N
E


E
)


g
(
T
,
p
)
s
SrTiO 3 SrTiO 3
TiO2 TiO2
O2 O2


2A 
2
Oxygen Gibbs free energy
 pO2 
gO2 ( pO2 , T )  gO2 ( p , T )  gO2 ( p , T  T )  kT ln 0 
 p 
0
0
1 0
1
0
0
g O2  gTiO2  gTi 
G 0f TiO2
2
NA
Experimental Value
0
0
We used 12 oxides: SrO,
TiO2, MgO, SiO2, Al2O3,
CaO, PbO2, CdO, SnO2,
Cu2O, Ag2O, ZnO
Thermodynamics of Surface Reconstructions
  pO , T , ETiO
2
2

1  0K
1

0K

(
E

N
E


E
)


g
(
T
,
p
)
s
SrTiO 3 SrTiO 3
TiO2 TiO2
O2 O2


2A 
2
0K
min
max
0K
ESrO
 ETiO

E

E

E
TiO2
TiO2
TiO2
2


1  0K
1

0K
0K
 pO2 , T 
(
E

N
E


E
)


g
(
T
,
p
)
s
SrTiO 3 SrTiO 3
TiO2 SrO
O
O

2 A 
2 2 2
Calculated from first principles
First principles +
analytical expression
The dependence of the surface energy with p and T comes through the gas phase.
Results: Kubo and Nozoye
~1200K
Coverage Θ
(1x1) Θ=1
As we increase the
temperature,  tends
to decrease (not
monotonically) as the
surface goes through
a sequence of
reconstructions.
(2x1) Θ=0.5
UHV=5x10-12 atm
~1500K
c(4x2) Θ=0.25
T. Kubo and H. Nozoye, Surface Science 542 (2003) 177-191
Results: Kubo and Nozoye

surf. energy: pO2 , T

min
0K
ETiO

E
TiO2
2
gOmin
 gO2 (T , p)
2
0: TiO2-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
~1200K
Equilibrium with SrO
4
~1500K
2
3
1
0: TiO2-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 .
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 SrTiO3 (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.