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1 Chapter 2. Structure A. Electronic structure vs. Geometric structure B. Clean surface vs. Adsorbate covered surface (substrate + overlayer) C. Adsorbate structure - how are the adsorbed molecules bound with respect to the substrate atoms? - how are they bound with respect to one another?

2 A) Crystal lattices (1) Lattices and unit cells A crystal is built up from regularly repeating structural motifs. Structural motifs may be atoms, molecules, or groups of atoms, molecules, or ions. Space lattice is a pattern formed by points representing the locations of structural motifs. - a 3D, infinite array of points, each of which is surrounded in an identical way by its neighbors, and which defines the basic structure of the crystal. * motif: a single or repeated design or color

3 ex) a molecule or a group of molecules * Crystal structure = collection of structural motifs arranged according to the lattice

4 Unit cell: imaginary parallelepiped (parallel sided figure) that contains one unit of the translationally repeating pattern ' [pæ rəlèləpáipid]

5 Unit cell

6 Normally chosen primitive unit cell with shortest lengths, which are most nearly perpendicular to one another

7 Cubic system Monoclinic system triclinic system

8 There are only 14 distinct space lattice in 3 dimensions. Bravais lattices P: primitive(with lattice points only at corners) I:body-centered (lattice points at corners & at its centers) F:face-centered(lattice points at corners & at centers of its 6 faces) (A,B,or C): side-centered (lattice points at corners & at the centers of 2 opposite faces)

9 (2) The identification of lattice planes (a) The Miller indices (hkl) - reciprocals of intersection distances (with fractions cleared by multiplying through by an appropriate factor)

10 (1a,1b) or (1,1), (1,1, ) (-1a, 1b)or (-1,1), (-1,1, ) (½ a, 1/3b) or (½, 1/3) (1/2, 1/3, ) ( a, 1b) or (, 1) (, 1, )

11 (1,1,0) Miller index

12 (2,3,0)

13 (1,1,0)

14 (0,1,0)

15

16 (b) The separation of planes Ex) separation of the {hkl} planes in the square lattice 1/d hkl2 = (h 2 +k 2 +l 2 )/a 2 Ex) general orthorhombic lattice 1/d hkl2 = h 2 /a 2 +k 2 /b 2 +l 2 /c 2 The dimensions of a unit cell and their relation to the plane passing through the lattice points

17 (3) Crystal structure (1) Metallic solids: electrons are delocalized over arrays of identical cations and bind them together into a rigid but ductile and malleable whole (a) Close packing The first layer of close-packed spheres to build a 3D close-packed structure

18 -The second layer of close-packed spheres occupies the dips of the 1st layer. The two layers are the AB component of the close-packed structure.

19 -polytypes: structures that are identical in 2D but differ in the third dimension. ABA (hcp) ABC (ccp:fcc)

20 (1) hcp(hexagonally close-packed) : ABABABAB (2) ccp(cubic close-packed): ABCABCABC.. (3) Coordination number = the # of atoms immediately surrounding any selected atom(=12) (4) Packing fraction = the fraction of space occupied by the spheres (=0.740) Calculation of packing fraction of ccp unit cell

21 Band Structure of Surfaces E E g ; bandgap = conduction band minimum valence band maximum Eg k

22 Formation of molecular orbital H2

23 N molecular orbitals covering a band of energies of finite width

24 When N electrons occupy a band of N orbitals, it is only half full and the electrons near the Fermi level(top of the filled levels) are mobile.

25 Fermi-Dirac Statistics Fermi function F(E) = 1/{1+exp[(E-E F )/k B T}} : probability that a certain energy level E is occupied by an electron (1) when an energy level E is occupied by an electron, F(E)=1 (2) For an empty level, F(E)=0 At T=0, F(E) = 1 for E<E F At T>0, some electrons near E F acquire enough energy to move into levels E>E F, emptying some of the levels immediately below E F. When E=E F, F(E) = ½ (generic definition of Fermi energy)

26

27 (a) When 2N electrons are present, band is full and it is an insulator at T=0 K (b) T>0, electrons populate the levels of the conduction band at the expense of the filled valence band semiconductor

28 * Intrinsic semiconductors : semiconductor whose electrical conductivity is dominated by the thermally generated EHPs n=p=n i, n=concentration of e, cm -3 & p=concentration of h & n i =intrinsic carrier concentration -carrier concentration is determined by (1) generation of carriers by thermal excitation (2) recombination of e and h due to energy considerations In a steady state (generation rate of EHPs g i = recombination rate of e and h r i ) g i = r i

29 * Extrinsic semiconductors Doping : introduction of charge carriers by intentional addition of impurities into the lattice Dopants : impurities - donors: have extra electrons (n-type semiconductor) - acceptors: have extra holes (p-type semiconductor)

30 Low T High T

31 (1) Elemental semiconductors: Si, Ge - n-type dopants: As, Sb, P - p-type dopants: B, Al, Ga, In (2) Compound semiconductors : GaAs, InP, CdTe, ZnSe - n-type or p-type dopants depends on the sublattice on which a particular impurity resides ex) Si :donor on Ga sublattice, acceptor on As lattice : experimentally amphoteric impurity (3) For III-V compound - n-type dopant: S, Se, Sn - p-type dopant: Cd, Zn, Mn, Be

32

33 * Carrier concentration - Influence of doping on the Fermi level

34 The position of the Fermi energy in a doped semiconductor depends on the concentration and type of dopants. (1) n-type semiconductor - E F is pushed upward E F = E i + k B T ln(n D /n i ) (2) p-type semiconductor - E F is pulled toward the valence band E F = E i k B T ln(n A /n i )

35 * Concentrations of electrons and holes at equilibrium concentration of electrons in conduction band n o = ECB N(E)dE, N(E)=# of e /unit energy within E and E+dE = N c F(E CB ) Probability of occupancy at E CB Effective density of states located at the conduction band edge E CB n o = N c exp{-(e CB -E F )/k B T} when (E CB -E F )>> k B T p o = N v exp{-(e F -E VB )/k B T} when (E F -E VB )>> k B T

36 In an intrinsic material, E F is at the same level E i in both cases. n i = N c exp{-(e CB -E i )/k B T} and p i = N v exp{-(e i -E VB )/k B T} n o p o = N c N v exp{-(e CB -E F )/k B T} exp{-(e F -E VB )/k B T} = N c N v exp{-(e CB -E VB )/k B T} = N c N v exp{-e g /k B T} since n i = p i, n 2 i = N c N v exp{-e g /k B T} n o p o = n 2 i

37 Friedel Oscillations: The electron density near the surface oscillates before decaying exponentially into vacuum.

38 Energy levels at metal interfaces -Interface: boundary between two materials in intimate contact -At equilibrium, the chemical potential must be uniform throughout a sample. = Fermi levels of two materials, which are both at equilibrium and in electrical contact, must be the same.

39 -When two metals are connected electrically, electrons flow from the low work-function metal to the high-work function metal (from L to R), until the Fermi levels become equal. -Metal L is slightly depleted of electrons and metal R has an excess of electrons: dipole develops between the two metals with a small potential drop and an electric field. -Contact potential, C = R - L (difference in work function)

40 Metal-semiconductor junctions (1) Rectifying junctions (Schottky barrier diodes): Metal/n-type semiconductor M : work function of metal s : work function of semiconductor : electron affinity of semiconductor (E vac E CB ) Assume M > s (E f (M) is at a lower position than E f (semiconductor))

41 (1)When junction is formed - Electrons from E CB of semiconductor flow to metal until E F are aligned. ( E CB is higher.) - Electron flow from n-type Si leaves behind positively charged donor ions in the thickness W. - Near the surface region, E F must move deeper into semiconductor, leading to upward bending of energy bands. - Electric field from n-side to metal. - Equilibrium contact potential of the junction V o prevents further flow of electrons from semiconductor to the metal. (V o = M - s ) - Barrier height q B for the injection of electrons from metal into semiconductor Schottky barrier q B = q M - q (cf, in case of p-type Si, M < s : Schottky junction)

42 (b) Rectifying behavior (1) At equilibrium, electrons having E>qV o are thermionically emitted over the barrier into the metal. (2) If heavily doped, W becomes thin. -Electrons can tunnel into metal through the barrier. -Current I MS (metal to semiconductor) -At equilibrium, net current =0, i.e., I MS = I SM

43 (2) With forward bias V f (n-type, negative) potential barrier to the flow of electron from semiconductor to metal decreases from qv o to q(v o -V f ) increase in I MS above equilibrium value. but I SM does not change since q B remains unaltered on biasing. net flow of current from metal to semiconductor occurs. (3) With reverse bias V r (n-type, positive) reduction of electron flow from semiconductor to metal due to increase in barrier height V o +V r I MS is reduced below equilibrium value, I SM remains almost the same. a small current flow. similar I-V behavior of Schottky barrier diode and p-n junction.

44

45 (2) Nonrectifying junctions (Ohmic contact) -metal/semiconductor contact has a negligible contact resistance relative to bulk or series resistance of semiconductor. (a) M < S for n-type semiconductor -Fermi level is aligned by transfer of electrons from metal to semiconductor. -Raises the semiconductor electron energies relative to that in the metal at equilibrium. -Barrier to the flow of electrons from metal to semiconductor is small and can be easily overcome by application of small voltage. (b) M > S for p-type semiconductor -Hole flow across the junction can occur easily. -Depletion regions are not formed because the alignment of Fermi levels requires the accumulation of majority carriers in the semiconductors.

46 I V

47 Work function depends both on the crystallographic orientation of the surface and on the presence of surface reconstruction. -Adsorbates on metal W(110) > W(100) > W(111), smallest Φ for least densely packed surface (1) Alkali metal on transition metal donates charge density to metal to decrease the work function. (2) O, S, halogens withdraw charge to increase the work function. (3) Molecular adsorbate with dipole - increase or decrease work function depending on the relative orientation of the molecular dipole with respect to the surface. - Increase in workfunction +

48

49 Heat Capacity A solid consisting of N A atoms has molar internal energy, U m : U m = 3N A kt = 3RT (N A k = R) Therefore, molar heat capacity, Experimental data C V,m = (du m /dt) V = 3R CONSTANT! (=24.9JK -1 ) However, it was found that C V,m 0 as T O Einstein introduced quantum concept to explain the heat capacity. Later, improved by Debye.

50 Vibrations of solids Einstein s assumption (1) Each atom oscillates about its equilibrium position with a single frequency (2) Energy of oscillation is confined to discrete values, nh Um = 3N A h / {e h /kt 1 }, C v,m = 3Rf 2, f = E /T {e E /2T / (e E /2T -1)}, E = h /k: Einstein temperature - at low T, only a few oscillators possess enough energy to oscillate but at high T, all N oscillators are excited and C v approaches its classical value!!!

51 Debye formula - atoms oscillate over a range of frequency, - frequency : D 0, D = h D /k : Debye temperature -: good agreement with experiment!!! - Since the surface atoms are undercoordinated compared with the bulk, surface Debye frequency is much lower than the bulk Debye frequency. - In particular, the vibrational amplitude perpendicular to the surface is much larger for surface than for bulk atoms.

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