5 . 1 Motivation
The physical and chemical propert ies of isolated, small clusters of t he semiconduct ing group 14 elements
(
Sin , Gen , Snn)
have been t he subject of intense research due to t hefundamental interest and the possibility of diverse applications in nano-technologies [ 1 27] .
The electric dipole moment and dipole polarizability are especially interesting propert ies to probe, because t hey provide bot h a characterizat ion of t he geometrical and the elec t ronic structure of t hese small nano-sized part icles. Particularly tin clusters have become t he center of focus due to t heir occurrence in bot h covalent and metallic bulk phases [128]
and t heir abnormal higher melting temperat ures as compared to t he bulk value [1 29,130].
The higher rat io of surface to bulk atoms was believed to always cause size-dependent melt ing temperat ure depression of nanoclusters [1 3 1-133] 1. However, SnlO was the first reported example of a nano-particle melting at a higher t emperature t han its bulk phase
and contradicted t he above ment ioned standard paradigm based on t hermodynamic argu ments. Melting properties measured by nanocalorimet ry and ion mobi lity measurements and calculated by density funct ional met hods and molecular dynamics simulations are reported in [ 1 29, 1 30, 1 34-138] .
In conclusion, significantly higher melt ing temperatures t han t hat of bulk tin (505 K ) were found for most o f t he t in clusters and were attributed t o t he covalent nat ure of bonding and to geometrical factors2 .
1 For instance, Au clusters with a radius of 3 nm ( around 700 atoms) melt at about 800 1< , compared to a bulk melting point of 1 337 K [132] .
2The reported melting temperature of 8nlO, for instance, is around 2000 K [136] .
78
CHAPTER 5. TIN CL USTERS 79
The normal allotrope of tin under ambient conditions (jJ-Sn, white tin) is a metal with a body-centered tetragonal lattice, opposed to its lighter congeners, Si and Ge which are already semiconductors at room temperature. Below 286 K , there is a stable semiconducting phase (a-Sn, grey tin) with a small band gap of 0 . 1 eV and a cuhic, diamond structure similar to Si and Ge.
Due to t he high relevance of silicon clust ers in t he nano-electronic industry, a vast
range of t heoretical invest igations into the geometric stable structures of neutral and
charged, small and mid-sized silicon clusters have been published [139-153]. Jackson et
al. [154] and Maroulis et al. [ 155] calculated t heir response to an external stat ic electric
field. The former group found the per atom polarizability of Si2o-28 for compact struc
t ures to decrease towards t he bulk limit , whereas pro late structures become increasingly p olarizable. Since jellium-based models for t he polarizability captured this trend nicely,
t he response to t he applied field is claimed to be metallic. The l atter group focused on
cluster sizes three to seven and found that , for t he different ial mean polarizability per
atom3, ab initio and density functional t heory based calculat ions yield distinctively dif
ferent pictures. Earl ier, Schiifer et al. measured t he polarizabilit ies o f Sin clusters by a
laser vaporation beam deflect ion met hod at a nozzle t emperat ure of 300 K (9 :S
n
:S 1 20)and found a very strong variation of t he polarizabilities per atom around t he bulk value of 3 . 7 1 A 3 [ 1 56 , 200] . These variat ions about t he bulk value were not found by J ackson et
at. [ 154] and have not been reported by t heoretical methods so far. A very revealing study
is t he spectroscopic evidence for the tricapped trigonal prism structure (TTP) , which is
a dominating structural motif in small, neutral Si, Ge and Sn clusters, for anionic silicon clusters [ 1 5 7] .
Germanium clusters have attracted similar theoretical and experimental interest com pared to silicon in the past two decades. They belong to t he most important m icro electronics materials and understanding t heir growth-habit and various electronic prop erties is of fundamental and substantial practical relevance. A vast range of t heoretical studies on the structures of neutral and charged germanium clusters up to 40 atoms have been reported [1 40, 1 58-1 64]. In general, the claimed ground-state structures were ob t ained using a genetic algorithm combined with t ight-binding methods or a basin-hopping algorithm coupled with plane-wave pseudopotential density functional calculations.
CHA PTER 5. CL USTERS 80
former group calculated t he polarizabilities of Gen
(
n = 2 -25) from B3LYP Lo -Alamos pseudopotential calculat ions using a valence double-zeta basis set(LA L2DZ)
and con cluded that in general, the polarizabilities per atom ineTease wit h increasing cluster ize. but show various fluct uat ions around 5.4 A 3In
for t he Clllst r siz 5 to 25. T h se fl uc t uations are claimed to cone pond t o clusters wit h large HO�IO-LUl\IO gap , resulting in smaller polarizabilitie . These are, however, ill cOllt rauict iouiJ wiL h t ho 'e of the latt er group. Chelikowsky et al. [165J invest igated t he polarizabilit ie of silicon, ger manium and germanium-ar enide clusters up to ten at oms and find t hese to deer'cascwit h increasing size towards t he bulk limit . They find ident ical g omet rical for
Sb- 10 and Ge2
-
10,
where the germanium cluster how an average increase of interatomicd istances by about 4 o/c , and exhibit about 10 1C high r polarizabilit ies per atom t han Si clusters. Ion mobility measur ments of germanium cluster ion r vealed t hat clust rs \vith about 1 0 to 40 atoms follow a one-dimensional growth-pattern to give prolate geomct ries, those with
40
to 70 atoms retain roughly t he same aspect ratio and at about70
atoms, the clusters reconstruct to a more spherical geometry [ 166]5 .Early st udies on t in cluster date back to 1953. Honig ident ified ionic clusters of up
to five t in atoms upon vaporizat ion of t in
[167] .
The atomization energies of t in clu terwith up to seven atoms and t he evaluation of t heir t ability under equilibrium condit ion
were invest igated by Gingerich et al. [16 J . Anderson calculated t he binding energies of
tin clu t ers wit h different geomet ric struct ures and compared them to the aforement ioned
t udy [ 169J . The energet ic separat ion and geometries of a variety of different neutral and positively charged tin cluster 6 (Sn2 > Sn3 , Sn.l) Snt , Sn5, Sn
t)
were reported by Balasubramanian et al. [1 70-1 75J.The ground state vibrat ional frequency and equilibrium molecular const ants of Sn2 were measured by �Iiller et al. [ 1 76] . l\Ieasured photoionizat ion and -emission spectra are reported in [1 77-179]. No regions of unusual high stability were observ d in t he mass spectra of cationic ilicon 7 ) germanium and t in clusters by Schaber et al. [180J . St udies on the react ion of t in clu ter anions wit h oxygen revealed t hat no oxide products were observed for c lusters larger than Sn5 [181J . Calculated dissociation energies, bond lengt hs
4 Although otherwise tated by Zhao et al., who unfort unately do not compare their result with the
bulk limit polarizability.
5 An indication for metallic clusters.
61n most cases respective germanium and lead cluster are also reported. 7The produced ilicon clusters contained a significant amount of hydrogen.
CHAPTER
5. TINCL USTERS
8]and harmonic frequencies of silicon, germanium and t in dimers are compared with ex perimental results in ref. [ 1 82] . J arrold et al. characterized t he struct ures of t in cluster
cations up to n = 68 by ion mobility measurements and summarized t hat up to n � 35,
tin cluster cations show a prolate growth-pattern [1 83]. They also probed structures of lead cluster cat ions up to n = 32 by t he same means and concluded that t hese clusters
adopt near-spherical geometries, indicating t he expected growth-pattern for metallic clus ters [ 1 84] . According to t hese findings, t he transition to » normal" metal cluster growth in t he group- 1 4 of singly positively charged clusters occurs between tin and lead . Lee
et al. calculated the ionization potent ials and binding energies of small germanium and t i n clusters with up to 13 atoms using a semi-empirical t ight binding met hod and found reasonable agreement with experimental results [185] . Much better agreement wit h ex perimental results for these properties were found by Maj umder et al. , who calculated the ground state geometries and energetics of neutral and singly positively charged t in clusters up to 20 atoms using the ultrasoft pseudopotential plane-wave method with gen eralized gradient approximations [186]. With respect to collision induced fragment ation processes of small tin cluster cations (Sn4-20) in t he energy range of 0-300 e V, Tai et al.
concluded t hat smaller clusters (n :s: 1 1 ) fragmented by t he atom loss process and t he l arger ones decayed by fission . Furt hermore, t hey found t hat t hese favored fragment ation paths resembled those for respective Si and Ge cluster ions, hence, confirming the struc t ur al similarities. They also backed their experimental results by t heoretical calculations utilizing various DFT methods [ 187, 1 88] .
By starting with h igh-symmetry structures and distorting them according to their
unstable modes , Pushpa et al. calculated low-energy structures of Mn (M = Sn, AI, As;
n = 4, 6,13) using DFT methods [189] . Alt hough t his is a conceptionally simple way
t o search for low-energy structures, and it also mimics t he way in which t he J ahn-Teller effect leads to particularly low-symmetry st ructures, it does not provide a proper route
to search for global minima. Maj umder et al. investigated t he structures, energetics and
fragmentation behavior of tin clusters up to 20 atoms by means of DFT methods and reported a different growth behavior to that of germanium and silicon clusters for tin clusters with more than 7 atoms [ 1 90] . A comparative t heoretical study with different exchange-correlation functionals of geometric structures and some electronic properties of
t i n clusters up to 20 atoms was published by Majumder et al. in 2005 [ 1 9 1] . A stable
CHAP TER 5. T IN CL USTERS
photoelectron spectl'oscoPY recently [1 92].
82
Considering the vast amount of theoret ical and experimental studies undertaken on small t in clusters, it become evident t hat neit her calculated , nor experimentally evalu ated static electric dipole polarizabilities have heen published. Furth rmore, all reported theoretical work on geometric structures was undertaken by using low-lying isomer struc t ures of silicon and germanium clusters as an illitial struct ure and then relaxing t hem into their local minima, rather than undertaking an unbiased search. Moreover, t he t heoretical approaches d here, invest igated only the lowest spin state of tin clusters. Th se three arguments are t he main motivat ion for t he present work.
5 . 2 Met hods
The predicted singlet and triplet global mll1lma ( G M ) struct ures o f t in clusters rang ing from seven up to twenty at oms were obt ained ut ilizing the genet ic a lgorithm code
B E L P H E G O R as described in det ail in section (3. 1 ) . The initial populations8 consisted
of randomly generated structures, predicted Lennard-Jones global minima and predicted low-lying minima st ructures from t in [ 190J and silicon clusters [147, 1 93] from t he litera t ure. The minimum energy difference 6i
V
was set to 0 . 0 1 eV, dmin and dmax parameters were fixed between 2.2 A and 3.8A,
respectively. The termination criteria was 1 50 mat ing and local minimization steps for clusters up to 1 2 atoms and 100 steps for the remainder. The mutat ion probability was varied between 10 % and and 20 %.In contr ast t o cesium and gold clusters, t he quest for the GM was performed in com binat ion with the ultrasoft pseudopotential plane-wave method, leaving 5s25p2 valence electrons, within t he local spin-density approximation (LDA) as implemented in t he VASP
program package [ 194]9 . The cut-off energy for t he plane-wave expansion was chosen to be 6 Ry. The clusters were placed in a cubic cell of side 16 A with periodic boundary conditions. D uring t he global opt imizat ion the cell was dynamically adapted, ensuring a dist ance of greater t han 8 A between t he clusters.
Typically, eight to ten of the t hus obtained energetically lowest-lying isomers were then fmt her relaxed from LANL20z basis set and pseudopot ntial calculat ions t o t heir local
minima as implemented in the G AUSSIAN03 program package [33 1 ] . Depending on the
8Typically ranging from 10 to 15 different structures.
9VASP makes a better compromise between a realistic treatment of the electronic structure and the
CHA P TER 5. TIN CL USTERS 83
energy distribution, two to four of the energet ically lowest- lying true minima structures
obtained by t he e means were t hen furt her optimi7.ed using t he extensive STUTTGART valence basis set together wit h t he energy-consistent relativistic pseudopotent ial for tin [195] . Finally, harmonic vibrat ional frequencies were computed to ensure t hat t he relaxed geometries are true minima on the potent ial energy surface. For t he exchange-correlation potential, the hybrid funct ional b3p86, according to the parameterizat ion suggested by Becke [196] and Perdew [ 1 12] , was applied in a self-con istent fashion as implemented in the GAUSS IAN03 program package. No symmetry constraints were applied during t he opt imizat ion proc dure.
The same approach was adopted to find t he best DFT functional reproducing t he mean static dipole polarizability as mentioned in the cases of gold and cesium clusters. At first , the polarizability of the ground st ate