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# Lecture XI Solid state dr hab. Ewa Popko

Lecture XI Solid state dr hab. Ewa Popko. Resistivity (Ωm) (295K). Resistivity (Ωm) (4K). Material. 10 -5. Potassium. 10 -12. “Pure”Metals. 10 -10. 2  10 -6. Copper. Semi-Conductors. Ge (pure). 5  10 2. 10 12. 10 14. Insulators. Diamond. 10 14. Polytetrafluoroethylene

## Lecture XI Solid state dr hab. Ewa Popko

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1. Lecture XI Solid state dr hab. Ewa Popko

2. Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) Material 10-5 Potassium 10-12 “Pure”Metals 10-10 2  10-6 Copper Semi-Conductors Ge (pure) 5  102 1012 1014 Insulators Diamond 1014 Polytetrafluoroethylene (P.T.F.E) 1020 1020 Metals and insulators Measured resistivities range over more than 30 orders of magnitude

3. Metals, insulators & semiconductors? 1020- At low temperatures all materials are insulators or metals. Diamond 1010- Resistivity (Ωm) Germanium Pure metals: resistivity increases rapidly with increasing temperature. 100 - Copper 10-10- 0 100 200 300 Temperature (K) Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

4. 0 V(r) E2 E1 E0 Increasing Binding Energy r Bound States in atoms Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. . The potential energy of an electron a distance r from a positively charge nucleus of charge q is

5. + + + + + Nuclear positions R Bound and “free” states in solids The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is Where n = 0, +/-1, +/-2 etc. This is shown as the black line in the figure. 0 V(r) E2 E1 E0 V(r) Solid V(r) lower in solid (work function). r 0

6. + + + + + Electron level similar to that of an isolated atom Energy Levels and Bands In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms. Lower binding electron states become bands of allowed states. We will find that only partially filled bands conduct Band of allowed energy states. E position

7. Band Theory The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory. Free electron model

8. Band Theory U(r) U(r) The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory. Free electron model: Neglect periodic potential & scattering (Pauli) Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

9. Energy band theory Solid state N~1023 atoms/cm3 2 atoms 6 atoms

10. Metal – energy band theory

11. The effects of temperature At a temperature T the probability that a state is occupied is given by the Fermi-Dirac function Fermi-Dirac function for a Fermi temperature TF =50,000K, about right for Copper. where μ is the chemical potential. For kBT << EF μ is almost exactly equal to EF. n(E)dE The finite temperature only changes the occupation of available electron states in a range ~kBT about EF.

12. Band theory ctd. To obtain the full band structure, we need to solve Schrödinger’s equation for the full lattice potential. This cannot be done exactly and various approximation schemes are used. We will introduce two very different models, the nearly free electron and tight binding models. We will continue to treat the electrons as independent, i.e. neglect the electron-electron interaction.

13. 0 Influence of the lattice periodicity In the free electron model, the allowed energy states are where for periodic boundary conditions nx , ny and ny positive or negative integers. L- crystal dimension Periodic potential Exact form of potential is complicated Has property V(r+ R) = V(r) where R = m1a + m2b + m3c where m1, m2, m3 are integers and a ,b ,c are the primitive lattice vectors. E

14. Tight Binding Approximation Tight Binding Model: construct wavefunction as a linear combination of atomic orbitalsof the atoms comprising the crystal. Where f(r)is a wavefunction of the isolated atom rj are the positions of the atom in the crystal.

15. + + + + + Nuclear positions a The tight binding approximation for s states Solution leads to the E(k) dependence!! 1D:

16. = 10 g = 1 E(k) -p/a p/a k [111] direction E(k) for a 3D lattice Simple cubic: nearest neighbour atoms at So E(k) =- a -2g(coskxa + coskya + coskza) Minimum E(k) =- a -6g for kx=ky=kz=0 Maximum E(k) =- a +6g for kx=ky=kz=+/-p/2 Bandwidth = Emav- Emin = 12g For k << p/a cos(kxx) ~ 1- (kxx)2/2 etc. E(k) ~ constant + (ak)2g/2 c.f. E = (hk)2/me Behave like free electrons with “effective mass” h/a2g

17. Band of allowed states Gap: no allowed states Band of allowed states Gap: no allowed states Band of allowed states Each atomic orbital leads to a band of allowed states in the solid

18. Discard for |k| > p/a Reduced Brillouin zone scheme The only independent values of k are those in the first Brillouin zone. Results of tight binding calculation

19. The number of states in a band Independent k-states in the first Brillouin zone, i.e. kx</a etc. Finite crystal: only discrete k-states allowed Monatomic simple cubic crystal, lattice constant a, and volume V. One allowed k state per volume (2)3/Vin k-space. Volume of first BZ is (2/a)3 Total number of allowed k-states in a band is therefore Precisely N allowed k-states i.e. 2N electron states (Pauli) per band This result is true for any lattice: each primitive unit cell contributes exactly one k-state to each band.

20. EF Metal due to overlapping bands Metals and insulators In full band containing 2N electrons all states within the first B. Z. are occupied. The sum of all the k-vectors in the band = 0. A partially filled band can carry current, a filled band cannot Insulators have an even integer number of electrons per primitive unit cell. With an even number of electrons per unit cell can still have metallic behaviour due to band overlap. Overlap in energy need not occur in the same k direction

21. Part Filled Band Empty Band Partially Filled Band Part Filled Band Energy Gap Full Band Energy Gap Full Band EF EF INSULATOR METAL METAL or SEMICONDUCTOR or SEMI-METAL

22. Insulator -energy band theory

23. Covalent bonding Atoms in group III, IV,V,&VI tend to form covalent bond Filling factor F.C.C :0.74 T. :0.34

24. Covalent bonding 3D 2D Crystals:C, Si, Ge Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons are anti-parallel ) Example: Carbon 1S2 2S2 2p2 C C Diamond: tetrahedron, cohesive energy 7.3eV

25. Covalent Bonding in Silicon • Silicon [Ne]3s23p2 has four electrons in its outermost shell • Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice • Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)

26. diamond • Each atom has two states1s twostates 2s, six states 2p, two states 3s, six states 3p and higher • forN atoms there are 2N states 1s, 2N states 2s, 6N states 2p, 2N states 3s and 6N states 3p • After atoms get closer, the states get spiltted. The splitting is biggest for the states 3s and 3p. These states mix up giving 8N states. • At the equilibrium distance this band splits into two bands separated by the energy gap Eg. The upper – conduction band- contains 4N states and the lower – valence band also 4N states.

27. semiconductors

28. Intrinsic conductivity ln(s) 1/T

29. Extrinsic conductivity – p – type semiconductor

30. Conductivity vs temperature ln(s) 1/T