Clusters at Finite Temperature

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@Si16 Al@Li10, Al2@Li10 C@Ga12 and C@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 N=8,20,40,58,92,138,� Icosahedra for N=13,55,147,�

12. GS geometry of Nan (n=39-62)

14. Electronic Effect - GS Geometry

15. Electronic Effect - Connectivity

16. Electronic Effect � Thermodynamics I

17. Electronic Effect � Thermodynamics II

18. The heat Capacities

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 Z=50 : 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 >> weakest in Sn. Failed semiconductor. A solid on the verge of becoming a metal.

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.

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 : N=30 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 30�50 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

24. Gallium Clusters: Heat Capacity

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 > 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.

30. Magic Melters have geometric origin

31. Coordination Number

32. The MSD : Ga30 and Ga31

33. Electron Localization Function

34. The Lindemann Index

35. The Heat Capacity : Ga30- Ga31

36. Size Sensitivity: Ga17 and Ga20

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

42. Higher Melting temperature in Ga46

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

49. Ground State Geometries of Au20

50. Distribution of shortest bonds (2.63 �)

51. Heat Capacity Curves of Au19 and Au20

52. Structural transition in Au19 from 650- 900~K

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

57. Experimental Heat capacities for Al clusters

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

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@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(M=Ti,Hf,Cr,Mo and W) ? abundances for n=15,16 ? Minimum electron affinity for n=16 3. In SinM, M=Ti,Zr, and Hf ? n=8-12 basketlike open structures - most favorable ? n=13-16 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

73. Si16 @ Ti vs. Si16 1.Nature of bonding ? Si16Ti ? at ?ELF=0.70 Localization charge on the hexagonal rings. (Covalent bonds) ? at ?ELF=0.55 Four Si atoms (tetrahedron) connected to ring. (metallic like) ? Si16 ? at ?ELF=0.75 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 doesn�t 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 @Li10-12 Al as impurity in Li Tiny Alloy!

78. Equilibrium Geometries

79. Charge Density

80. Thermodynamics

81. C@Al and C@Ga Impurity Induced effects Host Al13 ( Icosahedra *) And Ga13 ( Decahedra)

82. Al13 and Al12C : Geometries

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 .