Lecture X Solid state dr hab. Ewa Popko

Measured resistivities range over more than 30 orders of magnitude Material Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) “Pure” Metals Copper Semi- Conductors Ge (pure) 5  InsulatorsDiamond10 14 Polytetrafluoroethylene (P.T.F.E) Potassium 2  Metals and insulators

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

Bound States in atoms Electrons in isolated atoms occupy discrete allowed energy levels E 0, E 1, E 2 etc.. The potential energy of an electron a distance r from a positively charge nucleus of charge q is V(r) E2E1E0E2E1E0 r 0 Increasing Binding Energy

Bound and “free” states in solids V(r) E2E1E0E2E1E0 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. r R Nuclear positions V(r) lower in solid (work function). V(r) Solid

Energy Levels and Bands + E position Electron level similar to that of an isolated atom Band of allowed energy states. 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 Theory The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory. Free electron model

U(r) Neglect periodic potential & scattering (Pauli) Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb) 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:

Solid state N~10 23 atoms/cm 3 2 atoms 6 atoms Energy band theory

Metal – energy band theory

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

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.

Influence of the lattice periodicity In the free electron model, the allowed energy states are where for periodic boundary conditions n x, n y and n y positive or negative integers. L- crystal dimension 0 E Periodic potential Exact form of potential is complicated Has property V(r+ R) = V(r) where R = m 1 a + m 2 b + m 3 c where m 1, m 2, m 3 are integers and a,b,c are the primitive lattice vectors.

Tight Binding Approximation Tight Binding Model: construct wavefunction as a linear combination of atomic orbitals of the atoms comprising the crystal. Where  (r)  is a wavefunction of the isolated atom r j are the positions of the atom in the crystal.

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

E(k) for a 3D lattice Simple cubic: nearest neighbour atoms at SoE(k) =  2  (cosk x a + cosk y a + cosk z a) Minimum E(k) =  6  for k x =k y =k z =0 Maximum E(k) =  6  for k x =k y =k z =+/-  /2 Bandwidth = E mav - E min = 12  For k <<  a cos(k x x) ~ 1- (k x x) 2 /2 etc. E(k) ~ constant + (ak) 2  /2 c.f. E = (  k ) 2 /m e k [111] direction  /a  /a   E(k) Behave like free electrons with “effective mass”  /a 2 

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

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

The number of states in a band Independent k-states in the first Brillouin zone, i.e.  k x  <  /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 /V in 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.

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 EFEF Metal due to overlapping bands

Full Band Empty Band Energy Gap Full Band Partially Filled Band Energy Gap Part Filled Band EFEF INSULATORMETAL METAL or SEMICONDUCTORor SEMI-METAL EFEF

Insulator -energy band theory

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

Covalent bonding 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 1S 2 2S 2 2p 2 C C Diamond: tetrahedron, cohesive energy 7.3eV 3D 2D

Covalent Bonding in Silicon Silicon [Ne]3s 2 3p 2 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)

diamond

semiconductors

Intrinsic conductivity ln(  ) 1/T

ln(  ) Extrinsic conductivity – n – type semiconductor

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature ln(  ) 1/T

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