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

MSE Ohm's Law:  V = I R voltage drop (volts)resistance (Ohms) current (amps) Resistivity,  and Conductivity,  : --geometry-independent forms of Ohm's Law E: electric field intensity resistivity (Ohm-m) J: current density conductivity Resistance: ELECTRICAL CONDUCTION

MSE-512 Resistivity and Conductivity as charged particles mobility,  = Whereis the average velocity is the average distance between collisions, divided by the average time between collisions, t d

MSE-512 Electrical Conductivity given by: 11 # electrons/m 3 electron mobility # holes/m 3 hole mobility Concept of electrons and holes: CONDUCTION IN TERMS OF ELECTRON AND HOLE MIGRATION

MSE Room T values (Ohm-m) CONDUCTIVITY  COMPARISON

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.

MSE Metals: -- Thermal energy puts many electrons into a higher energy state. Energy States: -- the cases below for metals show that nearby energy states are accessible by thermal fluctuations. CONDUCTION & ELECTRON TRANSPORT

MSE Insulators: --Higher energy states not accessible due to gap. Semiconductors: --Higher energy states separated by a smaller gap. ENERGY STATES: INSULATORS AND SEMICONDUCTORS

MSE Data for Pure Silicon: --  increases with T --opposite to metals electrons can cross gap at higher T material Si Ge GaP CdS band gap (eV) PURE SEMICONDUCTORS: CONDUCTIVITY VS T

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. 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. Electron (-) and hold (+) pair generation represented b a broken bond in the crystal. Both carriers are mobile and can carry current.

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

MSE Intrinsic: # electrons = # holes (n = p) --case for pure Si Extrinsic: --n ≠ p --occurs when impurities are added with a different # valence electrons than the host (e.g., Si atoms) N-type Extrinsic: (n >> p) P-type Extrinsic: (p >> n) INTRINSIC VS EXTRINSIC CONDUCTION

MSE-512 Equations describing Intrinsic and Extrinsic conduction Using the Fermi-Dirac equation, we can find the number of charge carrier per unit volume as: N e = N o exp(-Eg/2kT)    is a preexponential function,    is the band-gap energy and  is Boltzman’s constant (8.62 x 10-5 eV/K  If Eg > ~2.5 eV the material is an insulator If 0 < Eg < ~2.5 eV the material is a semi-conductor Semi-conductor conductivity can be expressed by:  (T) =  o exp(-E*/nkT) E* is the relevant gap energy (Eg, Ec-Ed or Ea) n is 2 for intrinsic semi-conductivity and 1 for extrinsic semi- conductivity

MSE Data for Doped Silicon: --  increases doping --reason: imperfection sites lower the activation energy to produce mobile electrons. Comparison: intrinsic vs extrinsic conduction... --extrinsic doping level: /m 3 of a n-type donor impurity (such as P). --for T < 100K: "freeze-out" thermal energy insufficient to excite electrons. --for 150K < T < 450K: "extrinsic" --for T >> 450K: "intrinsic" DOPED SEMICON: CONDUCTIVITY VS T

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  = 1/  = q  d N d

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. E A and E D 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. Arsenic in silicon is qualitatively illustrated as a specific example (N D = cm -3 ). Note that at high temperatures ni becomes larger than doping and n≈n i. Devices are normally operated where n = N D +. Fabrication occurs as temperatures where n≈n i 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. Probability of an electron occupying a state. Fermi energy represents the energy at which the probability of occupancy is exactly ½.

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 mv 2 = ½ p 2 /m = h 2 /4  m k 2 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:

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

MSE Allows flow of electrons in one direction only (e.g., useful to convert alternating current to direct current. Processing: diffuse P into one side of a B-doped crystal. Results: --No applied potential: no net current flow. --Forward bias: carrier flow through p-type and n-type regions; holes and electrons recombine at p-n junction; current flows. --Reverse bias: carrier flow away from p-n junction; carrier conc. greatly reduced at junction; little current flow. P-N RECTIFYING JUNCTION

MSE-512 Piezoelectrics Field produced by stress: Strain produced by field: Elastic modulus:  = electric field  = applied stress E=Elastic modulus d = piezoelectric constant g = constant

2 Created by current through a coil: Relation for the applied magnetic field, H: applied magnetic field units = (ampere-turns/m) current APPLIED MAGNETIC FIELD

          

4 Measures the response of electrons to a magnetic field. Electrons produce magnetic moments: Net magnetic moment: --sum of moments from all electrons. Three types of response... Adapted from Fig. 20.4, Callister 6e. MAGNETIC SUSCEPTIBILITY

 

Hysteresis Loop

Soft and Hard Magnetic Materials

  

9 Information is stored by magnetizing material. recording head recording medium Head can... --apply magnetic field H & align domains (i.e., magnetize the medium). --detect a change in the magnetization of the medium. Two media types: --Particulate: needle-shaped  -Fe 2 O 3. +/- mag. moment along axis. (tape, floppy) --Thin film: CoPtCr or CoCrTa alloy. Domains are ~ 10-30nm! (hard drive) MAGNETIC STORAGE



MSE-512 Sheet Resistivity R = ====  s is the sheet resistivity Sheet resistivity is the resistivity divided by the thickness of the doped region, and is denoted  /□ L w If we know the area per square, the resistance is

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,  n and the strength of the field,  V d = -  n  The average drift velocity, v av is dependent Upon the mean time between collisions, 2 

Charge Flow and Current Density Current density, J, is the rate at which charges, cross any plane perpendicular to the flow direction. J = -nqv d = nq  n  n is the number of charges, and q is the charge (1.6 x C) OHM’s Law: V = IR Resistance, R(  ) is an extrinsic quantity. Resistivity,  (  m), is the corresponding intrinsic property.  = R*A/l Conductivity, , is the reciprocal of resistivity:  (  m) -1 = 1/  The total current density depends upon the total charge carriers, which can be ions, electrons, or holes J = q(n  n + p  p ) 