Conductivity, Energy Bands and Charge Carriers in Semiconductors MSE-630 Week 2 Conductivity, Energy Bands and Charge Carriers in Semiconductors

Objectives: To understand conduction, valence energy bands and how bandgaps are formed To understand the effects of doping in semiconductors To use Fermi-Dirac statistics to calculate conductivity and carrier concentrations To understand carrier mobility and how it is influenced by scattering To introduce the idea of “effective mass” To see how we can use Hall effect to determine carrier concentration and mobility

Conductivity Charge carriers follow a random path unless an external field is applied. Then, they acquire a drift velocity that is dependent upon their mobility, mn and the strength of the field, x Vd = -mn x The average drift velocity, vav is dependent Upon the mean time between collisions, 2t

Charge Flow and Current Density Current density, J, is the rate at which charges, cross any plane perpendicular to the flow direction. J = -nqvd = nqmnx = sx n is the number of charges, and q is the charge (1.6 x 10-19 C) The total current density depends upon the total charge carriers, which can be ions, electrons, or holes J = q(nmn + pmp) x OHM’s Law: V = IR Resistance, R(W) is an extrinsic quantity. Resistivity, r(Wm), is the corresponding intrinsic property. r = R*A/l Conductivity, s, is the reciprocal of resistivity: s(Wm)-1 = 1/r

As the distance between atoms decreases, the energy of each orbital must split, since according to Quantum Mechanics we cannot have two orbitals with the same energy. The splitting results in “bands” of electrons. The energy difference between the conduction and valence bands is the “gap energy” We must supply this much energy to elevate an electron from the valence band to the conduction band. If Eg is < 2eV, the material is a semiconductor.

Simple representation of silicon atoms bonded in a crystal Simple representation of silicon atoms bonded in a crystal. The dotted areas are covalent or shared electron bonds. The electronic structure of a single Si atom is shown conceptually on the right. The four outermost electrons are the valence electrons that participate in covalent bonds. Electron (-) and hold (+) pair generation represented b a broken bond in the crystal. Both carriers are mobile and can carry current. Portion of the periodic table relevant to semiconductor materials and doping. Elemental semiconductors are in column IV. Compound semiconductors are combinations of elements from columns III and V, or II and VI.

Doping of group IV semiconductors using elements from arsenic (As, V) or boron (B, III) Intrinsic carrier concentration vs. temperature.

Dopant designations and concentrations Resistivity as a function of charge mobility and number When we add carriers by doping, the number of additional carrers, Nd, far exceeds those in an intrinsic semiconductor, and we can treat conductivity as s = 1/r = qmdNd

Simple band and bond representations of pure silicon Simple band and bond representations of pure silicon. Bonded electrons lie at energy levels below Ev; free electrons are above Ec. The process of intrinsic carrier generation is illustrated in each model. Simple band and bond representations of doped silicon. EA and ED represent acceptor and donor energy levels, respectively. P- and N-type doping are illustrated in each model, using As as the donor and B as the acceptor

Behavior of free carrier concentration versus temperature Behavior of free carrier concentration versus temperature. Arsenic in silicon is qualitatively illustrated as a specific example (ND = 1015 cm-3). Note that at high temperatures ni becomes larger than 1015 doping and n≈ni. Devices are normally operated where n = ND+. Fabrication occurs as temperatures where n≈ni Probability of an electron occupying a state. Fermi energy represents the energy at which the probability of occupancy is exactly ½. Fermi level position in an undoped (left), N-type (center) and P-type (right) semiconductor. The dots represent free electrons, the open circles represent mobile holes.

The density of allowed states at an energy E. Integrating the product of the probability of occupancy with the density of allowed states gives the electron and hole populations in a semiconductor crystal.

Effective Mass In general, the curve of Energy vs. k is non-linear, with E increasing as k increases. E = ½ mv2 = ½ p2/m = h2/4pm k2 We can see that energy varies inversely with mass. Differentiating E wrt k twice, and solving for mass gives: Effective mass is significant because it affects charge carrier mobility, and must be considered when calculating carrier concentrations or momentum Effective mass and other semiconductor properties may be found in Appendix A-4

Substituting the results from the previous slide into the expression for the product of the number of holes and electrons gives us the equation above. Writing NC and NV as a function of ni and substituting gives the equation below for the number of holes and electrons:

The energy band gap gets smaller with increasing temperature. In general, the number of electron donors plus holes must equal the number of electron acceptors plus electrons The energy band gap gets smaller with increasing temperature. Fermi level position in the forbidden band for a given doping level as a function of temperature.

In reality, band structures are highly dependent upon crystal orientation. This image shows us that the lowest band gap in Si occurs along the [100] directions, whil for GaAs, it occurs in the [111]. This is why crystals are grown with specific orientations. The diagram showing the constant energy surface (3.10 (b)), shows us that the effective mass varies with direction. We can calculate average effective mass from: