Thermodynamics of clusters

1
Clusters at Finite Temperature
D.G. Kanhere Department of Physics and Centre
for Modeling and Simulation University of
Pune Pune 411 007
2
Quantum Dots

• Interacting electrons confinement
• Spin Density Function approach Configuration
  Interaction
• Bhalchandra Pujari and Kavita Joshi
• (POSTER)
• Couple Cluster Method (Singles and Double)
• Multireference Couple Cluster (excited states)
• Ideh Heidari and Saurav Pal (NCL)

3
Interaction of Aldehydes and Ketones with Gold
Clusters

• In collaboration with
• Ghazal Sadatshafaie
• Sharan Shetty
• See J. Chem. Phys. (Jan 2007)

4
Collaborators

• Kavita Joshi
• Sailaja Krishnamurty
• Mal-Soon Lee Sajeev Chacko
• Shahab Zorraisatein Steven Blundell
• Ghazal Sadatshefai Jarrold Martin
• Prachi Chandrachud

5
The talk is based on
6
The Outline

• Thermodynamics
• Homogeneous Clusters
• Sodium
• Gallium and Tin
• Aluminium
• Gold
• Impurity Doped Clusters
• Ti_at_Si16
• Al_at_Li10, Al2_at_Li10
• C_at_Ga12 and C_at_Al12

7
The Issues --

• Thermodynamics Beyond Ground State Properties.
  Exploring potential energy surface- Strong
  influence Of Ground state Geometry.
• No sharp transition due to finite size specific
  heat is broad.
• Melting Temperature could be ill defined
  melting over a broad range.
• Co existence region
• Solid like
• Liquid like
• And solid-liquid
• Pre-melting effects.
• Shell effects geometric and electronic.
• Isomerization effect of many low lying states.

8
Melting in Finite Size SystemsIssues

• The experimental observations
• Strong non monotonic variation in melting
  temperature Clusters. ( Sodium )
• Higher than Bulk melting point !! Tin and
  Gallium.
• Dramatic size sensitivity. Ga and Al Na and
  Gold Clusters
• The Ground State Geometry The Key
• Tuning finite temerature behaviour
• Effect of impurity, alloying ..

9
The story of Sodium clusters1997

• Large and irregular fluctuations observed in the
  melting temperature ( sizes 55- a few hundred).
• Tm is generally reduced w.r.t. bulk.
• All simulations /calculations ( mainly using
  interatomic potentials) yield Tm 100 K below
  experimental result.
• The peaks in the heat capacity do not correlate
  with either geometric or electronic shell
  closing.
• What About Electronic Structure effects?

10
Na continued

• Simplest of atomic clusters
• Jellium model works
• Nice delocalized charge density
• Magic Clusters at N8,20,40,58,92,138,
• Icosahedra for N13,55,147,

Chacko et al., Phys. Rev. B 71,155407 (2005) Lee
et al., J. Chem. Phys. 123,164310 (2005) Lee et
al., Phys. Rev. B (Accepted) Shahab et al., in
preparation
11
Experimentally measured Tm of Nan clusters
12
GS geometry of Nan (n39-62)
? Deformation parameter
13
Specific heat of Sodium clusters
14
Electronic Effect - GS Geometry

• Growth pattern of icosahedron
• Na58 (Electronic shell-closing system)
• nearly spherical shape without change
• of size compared to Na55
• compact GS geometry
• agree with experimental observation
• Ref. Issendorff and co-workers,
• Phys. Rev. Lett. 98, 043401
  (2007)

15
Electronic Effect - Connectivity
Side A
Side B
? Isosurface of ELF at 0.79 in Na58

• Connectivity of short
• bond lengths
• Na58 strongest but inhomogeneous
• Na57 intermediate between Na58
• and Na55
• Na55 weak but homogeneous

16
Electronic Effect Thermodynamics I
? Specific Heat

• Na58, Na57 (nearly) electronically closed
  shell with disordered structure
• high Tm, broad melting
  transition
• 2. Na55 geometric shell closing system
  ordered structure
• sharp melting transition

17
Electronic Effect Thermodynamics II
? Deformation Parameter
? After melting HOMO-LUMO gap closes
structure becomes elongated ? Nearly electronic
shell closing system, Na57 upon heating
structure becomes spherical from 180 K its
deformation parameter and eigenvalue
spectrum show similar behavior as seen in
Na58 (electronic shell closing)
? Time averaged Eigenvalue Spectrum
18
The heat Capacities
N40 peaked N50 flat N55 very sharp N58
peaked but broad
BUT Highest melting Point
( Tm 375 K )
19
Higher Than Bulk Melting Point

• Common Observation Tm for clusters is less than
  that of bulk goes as Tmelt(R) Tbulk (1 - C
  / R)
• Experiments by Jarrold (PRL) et al contradict
  this. The observed melting temperatures of Tin
  (Sn) and Gallium clusters in the range of 10-50
  atoms turn out to be at least 50 K above bulk for
  Tin and 200 -300 K more for Ga
• What is so special about Tin?
• Atomic Configuration Z50 5p2 ,5p2
• Group 14 C --- Si --- Ge --- Sn -- Pb
• Band Gap 5.5 1.2 0.7 0.1 Metal
  ( ev)
• Exists in diamond structure below 286K(
  semiconductor) and in BCT above ( Metal).
• Note Substantial Variation in the band Gap
  across the series indicates progressive weakening
  of sp3 hybridization gtgt weakest in Sn.
• Failed semiconductor. A solid on the verge of
  becoming a metal.

20

• Interesting features of Sn clusters
• Ground state geometries of small Sn, Ge, and Si
  clusters are very similar (Nlt20)
• HOMO-LUMO gap (seen through calculations or Sn
  1-1.5eV quite large
• The geometry of Sn10 is distinctly different than
  units seen in Solid State
• Support for this comes from photoelectron spectra
• Sn10 geometry is identical to Si10 and Ge10
• For N lt 30 geometries adopt prolate shape, again
  similar to Ge and Si
• Tricapped trigonal unit of nine atoms is seen as
  a building block up to N30
• Very large clusters are known to melt at
  temperatures below their bulk value
• It is reasonable to conjecture that the
  bonding in small Sn cluster is very different
  than bulk - covalent
• The behavior of Sn clusters should resemble
  that of Si /Ge

21
Fragmentation of Sn and Si

• The nature of bonding in Sn cluster is very
  different (covalent ) than that in bulk.
• Sn10 and Si10 show similar behavior.
• Small clusters of Sn and Si fragment and do not
  show liquid like behavior.
• In the case of tin clusters fragmentation occurs
  well above bulk Tm (505K)
• Our simulations indicate Sn20, Si15 and Si20 also
  fragment.
  

Krishnamurty et al., Phys. Rev. B 73, 45419
(2006)
Joshi
et al., Phys. Rev. B 67, 235413 (2003)

Joshi et al., Phys.
Rev. B 66, 155329 (2002)
22
The Story Of GalliumRecent Experiments by Breaux
et al PRL 91,215508(03)

• One more system with higher than bulk melting
  point Ga17, and Ga30 Ga55
• Size sensitive Heat Capacity N30 VS N 31
• Irregular variation in Tm Changes by 350 K
• Tm bulk is 303K
• Ab initio MD warranted
• 
  
  Chacko et al., PRL 126,8682 (04)

23
Ga Story ..Continues

• Recent experiments in the size range 3050 show
• Dramatic size sensitivity.
• Ga30 .. No peak , Flat Heat capacity
• Ga31 .. Shows a nice peak, well defined Tm
  
• Thus just a few atoms ( at times ONE!) make a
• difference

Joshi et al. Phys. Rev. Lett. 96,135703 (2006)
24
Gallium Clusters Heat Capacity
Experimental Data from Indiana group N30
Flat N31 Peaked Tm Shifts by 200K _at_ N45 And
by 350K across the series
25
The Analysis Tools
Electron Localization Function (A value of ELF
near 1 represents a perfect localization of the
valence charge density)
Mean Square Displacement
Diffusion coefficient
Velocity autocorrelation function
Root Mean Square Bond Length Fluctuations
(Lindemann Criteria)
Power spectrum
26
Treatment of dynamics Ab Initio MD

• Inter-atomic Potentials Electronics ions
• Classical Potentials
• Quantal effect
• Semi empirical (TB)
• Ab Initio Full Quantum mechanical treatment for
  electrons
• Method of simulation
  
  
• MC Monte Carlo
• Molecular Dynamics
• Time Scales?
• Classical MD gt 1 Nano sec
• Ab Initio ? 50 100 ps?

27
Calculations

• Density functional with LDA/GGA
• Born Oppenheimer MD
• Delta T 100 au
• Total simulation time per temp 100 ps or more
• Soft pseudo potentials, plane waves
• Extensive search for equilibrium geometries.
  Simulated annealing, Basin Hopping

28
The Multiple Histogram Method

• Finite temperature simulation of cluster
  generates discretely sampled configuration space
  points
• Statistical quantities like entropy and specific
  heat are desired
• The Ionic Density of States can be evaluated by
  via a fitting procedure of numerically measured
  probability distribution to analytical one
• For separable Hamiltonians, the configurational
  energy part can be separated and the probability
  of finding the cluster with configurational
  (potential) energy Vj is the central quantity.

29
Ref Gong et al., PRB v91, 14277 (1991)
30
Magic Melters have geometric origin
The Ground State Geometries
Ga31
Ga30
Joshi et al. Phys. Rev. Lett. 96,135703 (2006)
31
Coordination Number
For Ga30 most of the atoms have coordination
number 2 or 3 For Ga31 most of the atoms have
coordination number 3 or 4
32
The MSD Ga30 and Ga31
33
Electron Localization Function
a Ga30 b Ga31
34
The Lindemann Index
35
The Heat Capacity Ga30- Ga31
Joshi et al. Phys. Rev. Lett. 96,135703 (2006)
36
Size Sensitivity Ga17 and Ga20
Krishnamurty et al. Phys. Rev. B 73, 45406 (2006)
37
Geometric Origin of Size Sensitive Specific Heat

• Melters Ga31,Ga20,.. Na40, Na55
• Flat heat capacity Ga17, Ga30, Na50, Na58
• Ga31 more ordered ,has well defined planes
• Addition of one atom ( cap) displaces ALL atoms
  in Ga30 and changes co ordination
• Ga30 5 atoms with 4 fold or more co
  ordination
• Ga31 14 atoms with 4 fold or more NN
• Ga31 shows island of local order
• ELF identifies 26 atoms in such an island.

38
Gallium Clusters An analysis of the ground state
geometries in experimentally reported series
The atoms in white represent the extra atoms with
respect to earlier cluster size
Krishnamurty et. al. communicated (2007)
39
Growth pattern of Ga Clusters The clusters
undergo a spherical-prolate prolate-oblate Oblat
e-spherical transition spherical-prolate Transit
ion Ga46 has a very symmetric core
40
Symmetric distribution of the surface atoms
around the core for Ga46
41
ELF basins of various Gallium clusters
depicting The relative order in the clusters
42
Higher Melting temperature in Ga46
The strong network of core surface bonds in Ga46
lead to higher stability of the surface atoms
43
connectivity showing the core-core and
core-surface bonds leading to high tm
44
Finite Size Effect Amorphous Clusters

• In amorphous cluster each atom may have different
  environment.
• Atoms may be bonded with the rest of the cluster
  with different strength.
• When heated, they will begin diffusive motion at
  different temperatures
• This may result in continuous phase change.
• No sharp peak in specific heat, very broad
  transition.
• Cluster with large island having local order will
  show a peak in the heat capacity ..Most of the
  atoms melt together.

45
Specific Heat
46
Is This a generic phenomenon?

• Case of Gold clusters
• Sodium
• Aluminum

47
Gold Clusters Significance

• Recent reports revealed Au20 to be a tetrahedral
  
• structure with atomic packing close to that
  of bulk
• gold (FCC)
• Au19 is very similar to that of Au20 with one
  missing
• corner atom
• These two clusters are speculated to be highly
• stable with large HOMO-LUMO energy gap
• Proposed to be highly catalytically active
• A detailed thermodynamics of these two clusters
  will
• help in their applications
• J. Li, X. Li, H-J. Zhai, L-S. Wang, Science,
  2003, 299, 864

48
Geometries Au19
Continuous distribution of isomers
All low lying isomers are devoid of regular planes
Krishnamurty et. al. communicated (2007)
49
Ground State Geometries of Au20
High Energy Gap between the ground state and fist
low lying isomer
First low lying Isomer is a distorted
tertrahedron
50
Distribution of shortest bonds (2.63 Å)
Au20
Au19
Rest of the bonds are 2.75 Å in length
51
Heat Capacity Curves of Au19 and Au20
52
Structural transition in Au19 from 650- 900K
The Vertex Atom shifts from one edge to another
53
Au20 remains in its ground state Untill 700K
after which it undergoes A dramatic structural
transition leading to a sharp peak in the heat
capacity curve
54
Gold

• Gold is a case of Vacancy Induced diffusion
• leading to a rather flat heat capacity curve for
  Au19

55
Experimental Heat Capacities Al clusters
56
Variations in Melting Temperature
Al44 Sharp Peak Tm 600 K Al37 Broad
Peak Tm 850 K Al44 has 22 atoms with
coordination number 6 or more Al37 has 30 atoms
with coordination number of 6 or more
57
Experimental Heat capacities for Al clusters
Magic Melters 35 37 39 43 44 53 Non Melters
30-34, 49 Wide variation in melting
temperatures from 500K to 850K. Variation in
the shapes of heat capacity curves.
Starace et al. communicated (2007)
58
Simulations on Al

• Detailed DFT investigations of GS for 31, 34, 37,
  39, 40, 44, 46, 51.
• Detailed thermodynamic investigations of 37, 39,
  44, 46

59
Al Geometries

• 31 and 34 presence of voids
• 37 most compact structure
• 40 open structure with few defects
• 44 ordered GS
• 46 nearly degenerate GS

60
Shape Analysis

• ?def 1 similar growth along all the directions.
• 37 pyramidal
• 44,46 spherical
• 31, 34, 40 unequal growth along the principle
  directions.

61
Eigenvalue Spectra
37 and 46 show bunched eigenvalue spectra. Both
the clusters melt around 850 K.
62
Binding Energy per atom

• Binding Energy per atom shows local peak for Al37.

63
Shape sensitivity 34 Vs 44 Distance from COM

• Al44 well organized outer shell.
• Al34 continuous distribution of atoms from COM.

64
Coordination Number

• 44 32 out of 44 (80) atoms have 5/6 fold
  coordination. Most of the atoms are on the
  surface and experience similar environment.
• 34 12 atoms having 6 fold coordination are
  distributed throughout the cluster.

65
Variation in the melting temperature 37 Vs 44

• Coordination Number
• 37 80 of the atoms have 6 fold or more
  coordination
• 44 50 of the atoms have 6 fold or more
  coordination.

66

• 37 stronger core-surface connectivity.
• average coordination of core atoms is 9.7
• average bond distance 2.92 A
• 44 relatively weak core-surface
• connectivity. Average coordination is 7.2.
• average bond distance 3.3 A

67
Thermodynamics of Al clusters

• Al37

68

• Al39

69
Homogeneous Clusters Summary

• Na, Ga, Au, Al show size sensitivity
• Variation in the melting temperature
• Variation in the shape of the heat
  capacity curve
• Ga, Sn have higher than bulk melting
• temperature.
• Ground state geometry is THE KEY

70
Ti_at_Si16

• Pure Si clusters in this size range Fragment at
  1500K 1800 K
• Ti and other impurities have dramatic effect
• Formation of Si cage Vijay Kumar and co
  workers
• What is the effect of doping of such impurity on
  Finite temperature properties of Si?
• Thermodynamics!

71
Dopant induced stabilization of silicon clusters
at finite temperature

• 1. The medium silicon clusters in the size range
  of 15-20 atoms
• are unstable and fragment when heated up to
  approximately 1500 K.
• Is it possible to suppress such a
  fragmentation process ?
• 2. Experiments on doped silicon clusters
  SinM(MTi,Hf,Cr,Mo and W)
• ? abundances for n15,16
• ? Minimum electron affinity for n16
• 3. In SinM, MTi,Zr, and Hf
• ? n8-12 basketlike open structures - most
  favorable
• ? n13-16 the metal atom is completely
  surrounded by silicon atoms.
• 4. we show Ti-doped Si16 remains stable at least
  upto 2200 K and fragments only above 2600 K.

72
Gs geometry isomers of Si16Ti

• 1. Ground state
• ? Frank-Kasper polyhedron
• ?Two closely spaced shells
• . One with 4 Si atoms
• . Another with of 12 atoms
• 2. First isomer
• ? All Si atoms equidistant from Ti atom
• ? Energetically very closed to Gs (0.01 eV)
• 3. Isomers (Fig c-f)
• ? Distorted cage of Si atoms
• ? High energy compared to Gs
• ? Isomer (f)? seen at high temperature
• (2600 K )
• ? Possible path for fragmentation

Shahab et.al. Phys. Rev. B 75, 045177 (2007)
73
Si16 _at_ Ti vs. Si16

• 1.Nature of bonding
• ? Si16Ti
• ? at ?ELF0.70
• Localization charge on the
• hexagonal rings. (Covalent bonds)
• ? at ?ELF0.55
• Four Si atoms (tetrahedron)
• connected to ring. (metallic like)
• ? Si16
• ? at ?ELF0.75
• All Si atoms are connected via a
• single basin? bond is stronger in
• Si16 compared to Si16Ti .

74
Thermodynamics
75

• 1. Heat capacity
• ? Characteristic of finite size system
• ? Broad melting peak in 2250K ?Complete break
  down of Si cage
• 2. drms
• ? Si-Si bond ? high value at low temperature
• ?Si atom are mobile 600 to 1800K)?cluster
  partially melted
• ? Si-Ti bond ? rise after 2000K ?breaking of
  the Si cage and
• diffusing of Ti through out the cluster
• 3. g(r)
• ? width of second peak doesnt change
• significantly till 1600k?confined motion
  around Ti
• ? Continuous distribution after 2200K?Si cage
• is destroyed and Ti has escaped from the
  cage

76
Thus

• 1.The role of impurity in suppressing
  fragmentation Si16
• ? In Contrast to pure Si clusters, the doped
  cluster undergoes a solid like to liquid like
  transition, remaining stable at least up to 2600
  K.
• 2.Melting occurs in two steps
• ? First step is initiated by the surface
  melting at approximately 600K
• where Si atoms diffuse on the shell.
• ? Second step the destruction of the cage
  takes place at approximately
• 2250 K giving rise to a peak in heat
  capacity curve.
• 3.The present work demonstrate a possible way of
  tuning the finite
• temperature behavior of cluster using the
  appropriate dopant.

77
Al _at_Li10-12

• Al as impurity in Li Tiny Alloy!

78
Equilibrium Geometries
Mal-Soon Lee et al. Phys. Rev. B
79
Charge Density
Li10Al
Li10Al2
Difference charge density (Gain)
Difference charge density (Loss)
Total charge density
Li10Al localized around Al delocalized charge
distribution Li10Al2 more localized charge
distribution around Al atoms
80
Thermodynamics

• Isomerization shoulder
• Impurity effect
• Li10Al
• weakening of metallic bond
• around Al atom by charge
• transfer
• Premelting Feature at low
• temperature
• Li10Al2
• Al2 dimer-like behavior
• motions of Li atoms are
• confined around Al dimer
• NO melting until 800 K

81
C_at_Al and C_at_Ga

• Impurity Induced effects
• Host Al13 ( Icosahedra )
• And Ga13 ( Decahedra)

82
Al13 and Al12C Geometries
Two shell Icosahedra
Perfect Icosahedra
83
Al13 and Al12C Heat Capacities
84
Al13 and Al12C Delta
85
Ga13 and Ga12C

• Ga13 is Decahedra
• Ga12C is a Perfect Icosahedra

86
Ga13 and Ga12C Heat Capacities
87
Ga12C Delta
88
Summary

• Clusters at finite temperatures behave very
  differently than Bulk
• Melting points can be higher than bulk
• The shapes of the specific heat curves show
  dramatic size sensitivity.
• The nature of the ground state geometry dictates
  their finite temperature properties.
• Impurities can be used to TUNE their properties .