1. ENG2000: R.I. Hornsey Crystal: 1 ENG2000 Chapter 3 Crystals

2. ENG2000: R.I. Hornsey Crystal: 2 Overview of chapter In this chapter we seek to understand the types of crystal structures and their properties We also need to describe different directions and planes in crystals because the properties can be different in each direction There’s no such thing as a ‘perfect’ crystal, so we will look at how imperfections occur Later, we will build on these ideas when we look at material properties  semiconductors, magnetism, optical properties etc.

3. ENG2000: R.I. Hornsey Crystal: 3 Crystalline Solids A crystal is a material in which the atoms possess perfect ‘long-range order’  i.e. a repeating or periodic array of infinite dimension  this array is three-dimensional  for materials which crystallise, the crystal represents the minimum overall bonding energy of the system Crystals have well-defined chemical, physical and electronic properties  theoretically simpler  uniform and predictable properties  some properties are unique to crystalline form Generally, metals have the simplest crystal forms

4. ENG2000: R.I. Hornsey Crystal: 4 Unit cell All crystals comprise a fundamental, repeating block of atoms  this is called the ‘unit cell’  for most materials the unit cell is a parallelepiped with three sets of parallel faces  the entire crystal structure can be constructed from repeated translations in 3-D of the unit cell Several unit cells may be possible for a given crystal  the simplest and most symmetric is usually used  with atoms at the corners of the cell Callister hard sphere reduced sphere

5. ENG2000: R.I. Hornsey Crystal: 5 FCC Many common metals display the face-centred cubic (FCC) structure  Cu, Al, Ag, Au In the hard sphere representation, the atom cores on each face touch each other  hence the unit cell dimension, a, is given by a = 2R√2  where R is the core diameter a

6. ENG2000: R.I. Hornsey Crystal: 6 How many atoms in a unit cell? We have to be careful not to count atoms more than once  in FCC the corner atoms are divided between 8 neighbouring unit cells, so only 1/8 of each corner atom is in any one cell  but the face atoms are shared between only 2 unit cells So the total number is  (8 X 1/8) corners + (6 x 1/2) face = 4 atoms

7. ENG2000: R.I. Hornsey Crystal: 7 Other metrics The coordination of an atom (or coordination number) is the number of other atoms to which it is bonded  in FCC, this is 12  Si has a coordination of 4  this metric is especially useful when discussing mixtures or non-crystalline materials The atomic packing factor (APF)  APF = (total sphere volume)/(unit cell volume)  for FCC, the APF is 0.74  this is the largest possible for identical spherical atoms

8. ENG2000: R.I. Hornsey Crystal: 8 BCC Body-centred cubic (BCC) is found in materials such as W, Cr, and Fe For the hard (touching) sphere representation  a = 4R/√3  APF = 0.68 For BCC materials  the number of atoms in the unit cell is (8 x 1/8) + 1 = 2  and the coordination is 8  note that APF and coordination are related

9. ENG2000: R.I. Hornsey Crystal: 9 HCP Hexagonal close- packed (HCP) is found in Mg, Ti, Zn  coordination = 12  APF = 0.74  same as FCC Now there are 6 atoms in the unit cell  prove it! the unit cell a c http://www.usc.edu/dept/materials_science/MASC110L/hcp.jpg

10. ENG2000: R.I. Hornsey Crystal: 10 Comparison of metals metalstructureAtomic Radius (Å)* aluminumFCC1.431 chromiumBCC1.249 copperFCC1.278 goldFCC1.442 ironBCC1.241 leadFCC1.750 nickelFCC1.246 silverFCC1.445 titaniumHCP1.445 tungstenBCC1.371 * 1 angstrom (Å) - 10 -10 m = 0.1nm

11. ENG2000: R.I. Hornsey Crystal: 11 Silicon Unit Cell Unit cells can contain even more atoms  silicon has 8  one especially important consequence of more complex unit cells is that the density of atoms on a surface (and hence surface properties) depends on how the surface cuts through the unit cell  we need to be able to describe these planes – coming soon http://www.physics.monash.edu.au/~adamf/images/silicon.gif

12. ENG2000: R.I. Hornsey Crystal: 12 Seven crystal systems can be defined according to their lattice parameters x y z a b c    Callister

13. ENG2000: R.I. Hornsey Crystal: 13 Point coordinates In order to describe the directions and planes in a crystal, a set of coordinates has been developed  the coordinates of P are qrs (no commas), where q, r, and s are <1 x y z a b c P qa rb sc

14. ENG2000: R.I. Hornsey Crystal: 14 Crystallographic directions A direction is a vector between two points. Vectors should  pass through the origin (but can be translated without change)  the length of the vector projected onto the axes is determined in terms of a, b, and c  these numbers are reduced to the smallest integer values by multiplying or dividing by a common factor (also in units of a, b, c)  these three values are given as [uvw] [111] [???] [100] x y z e.g. [???]: vector is 0.5a1b0c multiply through by 2 x (a, b, or c) gives [120]

15. ENG2000: R.I. Hornsey Crystal: 15 Other directions e.g. Negative directions are indicated  also is in the opposite (antiparallel) direction to In a particular structure, more than one direction may have identical structures  e.g. cubic crystals  these is a family of directions, written as http://python.rice.edu/~arb/Courses/Images/360dot.gif

16. ENG2000: R.I. Hornsey Crystal: 16 Crystallographic planes Lastly, we can describe planes in a similar fashion  using (hkl), also called the Miller indices The procedure is as follows  the plane should not pass through the origin; if it does, either translate the plane or chose a new origin  the plane now either intercepts or is parallel with all the axes; the length of the intercept is determined in multiples of abc  the reciprocal of these multiples is taken (no intercept gives and index of 0)  these indices are reduced by multiplication or division by a common parameter (in units of abc) to their lowest integers  these are written (hkl)

17. ENG2000: R.I. Hornsey Crystal: 17 Callister

18. ENG2000: R.I. Hornsey Crystal: 18 Si [111] http://www.mse.nthu.edu.tw/jimages/Beuty/Si(111)-7x7%20.jpg

19. ENG2000: R.I. Hornsey Crystal: 19 Single crystals Perfect single crystals are hard to form  because impurities or defects are tough to prevent  single crystal metals – because of the lack of imperfections – are closest to the ideal mechanical strength Single crystals are fundamental to the semiconductor industry  they are drawn from a crucible of molten Si using a ‘seed’ crystal as a template  the growth rate is typically 1-10µm per second and the final ingot is about 1.5m long and up to 300mm in diameter  the ingot is then trimmed to a circular cross-section and sliced into wafers, which are then polished  imperfections in the crystal are measured per cm 2, a typical value being about 10 defects/cm 2 http://www.csc.fi/elmer/examples/czmeltflow/growth.gif http://www.ami.bolton.ac.uk/courseware/mdesign/ch2/SingleCrystalSiliconIngot.jpg

20. ENG2000: R.I. Hornsey Crystal: 20 Polycrystalline materials As they solidify naturally from the molten state, materials tend to become polycrystalline  consisting of many crystal grains, each with a random orientation, joining at grain boundaries  this results from the simultaneous growth, and subsequent coalescence, of crystals growing from multiple starting points http://mimp.mems.cmu.edu/~ordofmag/alumina.jpg http://www.mse.nthu.edu.tw/jimages/Beuty/Steel1.jpg Crystal grains of aluminum oxide ceramic Sheet steel

21. ENG2000: R.I. Hornsey Crystal: 21 Amorphous materials In contrast to crystals, which have perfect long-range order, amorphous materials have no long-range order Locally, the Si atoms still bond to 4 neighbours  but the bond lengths and angles vary randomly about the ideal values  so after >100 inter-atomic distances, the order is lost Amorphous materials are effectively ‘frozen liquids’  obtained if a liquid is cooled too rapidly to allow crystal formation http://www.research.ibm.com/amorphous/figure1.gif the continuous random network of amorphous silicon

22. ENG2000: R.I. Hornsey Crystal: 22 Applications Both amorphous and polycrystalline semiconductors find applications in electronics Single crystals must be formed from a single ‘seed’, and so cannot be formed on other substrates (e.g. glass)  so large-area devices, such as active matrix LCD displays, must be constructed from amorphous silicon  polycrystalline Si has better electrical properties but requires higher temperatures to form which distorts the glass

23. ENG2000: R.I. Hornsey Crystal: 23 Imperfections in solids As we mentioned before, effectively all crystals include imperfections  these can dominate the properties of the material in both desirable and undesirable ways The addition of impurities (i.e. other substances) is vital in metallurgy and microelectronics  e.g. sterling silver = 92.5% silver + 7.5% copper is much harder than pure silver  addition of B or P to Si drastically alters the Si electrical properties  we will get to these later in the course For the moment, we will discuss physical defects

24. ENG2000: R.I. Hornsey Crystal: 24 vacancy substitution self-interstitial (low probability in metals) Point defects Vacancies – missing atoms – are present in all materials  the number is given by N V = N exp (-Q V /kT)  where N is a constant, Q V is the energy required to create a vacancy, k is Boltzmann’s constant (1.38 x 10 -23 J/K) and T is the absolute temperature  for a metal just below melting, there is 1 vacancy for every ~10 4 atoms interstitial

25. ENG2000: R.I. Hornsey Crystal: 25 Impurities

26. ENG2000: R.I. Hornsey Crystal: 26 Impurities Because of the relatively ‘free-and-easy’ bonding structure of metals, mixtures of elements – alloys – are straightforward  the maximum purity achievable is ~99.9999%, or 1 in a million atoms is foreign An alloy is effectively a solid solution  the solvent is the species with the highest concentration  the solute is the lower concentration element There are two possibilities for forming the solution  substituting one atom for another  the solute atom fits in the interstitial site

27. ENG2000: R.I. Hornsey Crystal: 27 Substitution Solubility for substitutional impurities depends on a number of factors  relative sizes of atoms - typically limited to ±15% for high solubility  crystal structure – should be similar for high solubility  electronegativity – should be similar for high solubility  valence – a metal dissolves easier in another metal of lower valency that higher valency Copper/nickel is the example system where there is excellent solubility  R Ni = 1.25Å, R Cu = 1.28Å  both are FCC  electronegativities are almost equal  valence for Ni is +2, for Cu is +1

28. ENG2000: R.I. Hornsey Crystal: 28 Interstitial Since the metal packing densities are relatively high, the interstices are small  so only small atoms can dissolve in this way  even then, typically only 10% impurities can be added before the strain induced is too high Carbon is an interstitial impurity in iron  up to about 2%  R Fe = 1.24Å, R C = 0.71Å

29. ENG2000: R.I. Hornsey Crystal: 29 Specification of composition The composition of an allow can be specified in two principal ways Weight percent (wt%)  wt% 1 in 2 = m 1 /(m 1 + m 2 ) x 100  where m is the mass of each element Atom percent (at%)  at% is the number of moles of one element as a fraction of the total number of moles  the number of moles of material 1 is n m1 = m’ 1 /A 1, where m’ is the mass (in g) and A 1 is the atomic weight for material 1  at% 1 = n m1 /(n m1 + n m2 ) x 100

30. ENG2000: R.I. Hornsey Crystal: 30 ‘Mechanical’ defects

31. ENG2000: R.I. Hornsey Crystal: 31 Edge dislocations - linear defects An edge dislocation occurs when there is an extra crystal plane http://pilot.mse.nthu.edu.tw/tem/gallery/Tem-11.JPG h ttp://www.mse.nthu.edu.tw/jimages/Beuty/ copper sulphide cactus!

32. ENG2000: R.I. Hornsey Crystal: 32 http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/Burgers_circuit.jpg Burgers vector The direction and magnitude of a dislocation is expressed in terms of the ‘burgers vector’  “If you imagine going around the dislocation line, and exactly going back as many atoms in each direction as you have gone forward, you will not come back to the same atom where you have started  the Burgers vector points from start atom to the end atom of your journey”  for the edge dislocation here, the Burgers vector is perpendicular to the dislocation line

33. ENG2000: R.I. Hornsey Crystal: 33 http://www.uet.edu.pk/dmems/EdgeDislocation.gif

34. ENG2000: R.I. Hornsey Crystal: 34 Screw dislocation In screw dislocations, the atom planes look like they have been ‘sheared’ The Burgers vector is parallel to the line of the dislocation 350Å GaN http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/screw_disl_schem.gif http://nano.phys.uwm.edu/li/new_pa4.jpg Burgers vector

35. ENG2000: R.I. Hornsey Crystal: 35 Interfacial defects It is worth noting that any surface or interface is an imperfection  surface – dangling bonds that would otherwise have been occupied with other atoms lead to non-bulk electronic and mechanical effects at the surface (similar to surface tension in liquids)  grain boundaries  atomic vibrations – only a perfect crystal at 0 kelvin!

36. ENG2000: R.I. Hornsey Crystal: 36 How do we ‘see’ atoms? One cannot observe anything smaller than the wavelength of the illumination  ~500nm for visible light So how do we see atoms of size ~ 0.1nm?  not with light – this is in the x-ray part of the EM spectrum One possibility is an electron microscope  electrons have a wavelength that is inversely proportional to their energy, which depends on the acceleration voltage  energies in the range MeV are possible – what wavelength doe this correspond to? Another possibility is the scanning tunnelling microscope …

37. ENG2000: R.I. Hornsey Crystal: 37 http://jmaps.d.umn.edu/images/stm/stm1.gif For a small insulating gap, a current can flow because of the electron probability function. If the gap is small enough, there is a finite possibility that the electron is transmitted to the other side of the gap. This is called tunnelling.

38. ENG2000: R.I. Hornsey Crystal: 38 Summary Unit cells  FCC, BCC, HCP Coordinates, directions and planes  Miller indices Polycrystalline and amorphous materials Impurities  solid solutions  substitutions, interstitial Dislocations  edge, screw, Burgers vector