Diffusion & Crystal Structure

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1 Lecture 5 Diffusion & Crystal Structure

2 Diffusion of an interstitial impurity atom in a crystal from one void to a neighboring void. The impurity atom at position A must posses an energy E to push the The impurity atom at position A must posses an energy E A to push the host atoms away and move into the neighboring void at B.

3 Rate for a Thermally Activated t Process = Av exp( E A /kt) E A = U A* U A = frequency of jumps A = a dimensionless constant that has only a weak temperature dependence v o = vibrational frequency E A = activation energy k = Boltzmann constant, T = temperature, U A* = potential energy at the activated state A*, U A = potential energy at state A.

4 An impurity atom has four site choices for diffusion to a neighboring interstitial interstitial vacancy. After N jumps, the impurity atom would have been displaced from the original position at O.

5 Mean Square Displacement L 2 = a 2 t = 2Dt L = distance diffused after time t, a = closest void to void separation (jump distance), = frequency of jumps, t = time, D = diffusion coefficient Diffusion coefficient is thermally activated D a Do exp EA kt D = diffusion coefficient, D O = constant, E A = activation energy, k = Boltzmann constant, T = temperature

6 Example: Diffusion of P atoms in Si (to make n-type Si) To dope a Si wafer to make a PN junction, ions are usually implanted via a high energy accelerator ( kev). Then, the wafer is annealed to even the distribution of ions and get rid of any impurities. If the aneel is done at 1100 o C, what is the rms distance diffused by P atoms in 5 minutes? Take D o =10.5cm 2 /s and E A =3.69eV D EA 13 2 Do exp 3x10 cm / s L2 = 2Dt L = 13 μm kt Assuming the distance between voids in Si is ~2.7 Angstroms, the rate of jumps a P atom makes is: = 2D/a 2 = 823 jumps/sec

7 Crystal Structures Left: Galena is lead sulfide, PbS, and has a cubic crystal structure Right: Cubic FeS 2, iron sulfide, or pyrite, crystals. The crystals look brasslike ( fool s gold ).

8 Crystal Structures Left: Opals are periodic, close-packed dielectric spheres with diameters on the order of the wavelength of light Ri ht t t l l i t i f i di f ti i d Right: structural color in nature arises from periodic refractive index differences, also on the order of the wavelength

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10 Face centered centered cubic (a) The crystal structure of copper is face centered cubic (FCC). The atoms are positioned at well defined sites arranged periodically and there is a long range order in the crystal. (b) An FCC unit cell with closed packed spheres. (c) Reduced sphere representation of the unitcell. Examples: Ag, Al,Au,Ca, Au, Cu, γ Fe (>912 C), Ni, Pd, Pt, Rh.

11 Body centered cubic Example: Alkali metals (Li, Na, K, Rb), Cr, Mo, W, Mn, α Fe (< 912 C), β Ti (> 882 C) (a) A BCC unit cell with closely packed hard spheres representing the Fe atoms. (b) A reduced sphere unit cell.

12 Hexagonally close packed (a) The Hexagonal Close Packed (HCP) Structure. A collection of many Zn atoms. Color difference distinguishes layers (stacks). (b) The stacking sequence of closely packed layers is ABAB (c) A unit cell with reduced spheres (d) The smallest unit cell with reduced spheres.

13 Diamond Crystal lstructuret The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α Sn) and the Elemental semiconductors Ge and Si have this crystal structure.

14 Zinc Blende The Zinc blende (ZnS) cubic crystal structure. Many important compound crystal ( ) y y p p y Structures have the zinc blende structure. Examples: AlAs, GaAs, Gap, GaSb, InAs, InP, InSb, ZnS, ZnTe.

15 Rock kslt Salt Packing of coins on a table top to build a two dimensional crystal

16 A possible reduced sphere unit cell for the NaCl (rock salt) crystal. An alternative Unit cell may have Na + and Cl interchanged. Examples: AgCl, CaO, CsF, LiF, LiCl, NaF, NaCl, KF, KCl, MgO.

17 A possible reduced sphere unit cell for the CsCl crystal. An alternative unit cell may have Cs + and Cl interchanged. Examples: CsCl, CsBr, CsI, TlCl, TlBr, TlI.

18 Allotropy in Carbon

19 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw Hill, 2005)

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22 Labeling of crystal planes and typical examples in the cubic lattice

23 Kate Nichols (local artist and TED fellow): Exploring structural color

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