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
Conductivity, Energy Bands and Charge Carriers in Semiconductors
• Ohm's Law:
DV = I R
voltage drop (volts)
• Resistivity, r and Conductivity, s:
--geometry-independent forms of Ohm's Law
J: current density
Resistivity and Conductivity as charged particles
mobility, m =
is the average velocity
is the average distance between collisions,
divided by the average time between collisions,
CONDUCTION IN TERMS OF ELECTRON AND HOLE MIGRATION
• Concept of electrons and holes:
• Electrical Conductivity given by:
• Room T values (Ohm-m)
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.
CONDUCTION & ELECTRON TRANSPORT
-- Thermal energy puts
many electrons into
a higher energy state.
• Energy States:
-- the cases below
for metals show
ENERGY STATES: INSULATORS AND SEMICONDUCTORS
--Higher energy states
separated by a smaller gap.
--Higher energy states not
accessible due to gap.
PURE SEMICONDUCTORS: CONDUCTIVITY VS T
• Data for Pure Silicon:
--s increases with T
--opposite to metals
band gap (eV)
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.
INTRINSIC VS EXTRINSIC CONDUCTION
# electrons = # holes (n = p)
--case for pure Si
--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)
Equations describing Intrinsic and Extrinsic conduction
DOPED SEMICON: CONDUCTIVITY VS T
• Data for Doped Silicon:
--s increases doping
--reason: imperfection sites
lower the activation energy to
produce mobile electrons.
• Comparison:intrinsic vs
--extrinsic doping level:
1021/m3 of a n-type donor
impurity (such as P).
--for T < 100K: "freeze-out"
thermal energy insufficient to
--for 150K < T < 450K: "extrinsic"
--for T >> 450K: "intrinsic"
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
Dopant designations and concentrations
Resistivity as a function of charge mobility and number
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. 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.
Integrating the product of the probability of occupancy with the density of allowed states gives the electron and hole populations in a semiconductor crystal.
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:
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  directions, while for GaAs, it occurs in the . 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:
P-N RECTIFYING JUNCTION
• 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.
--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.
• Created by current through a coil:
• Relation for the applied magnetic field, H:
applied magnetic field
units = (ampere-turns/m)
mo=1.257 x 10-6 Wb/(A-m)
• Measures the response of electrons to a magnetic
• Electrons produce magnetic moments:
Adapted from Fig. 20.4, Callister 6e.
• Net magnetic moment:
--sum of moments from all electrons.
• Three types of response...
• Information is stored by magnetizing material.
• Head can...
--apply magnetic field H &
align domains (i.e.,
magnetize the medium).
--detect a change in the
magnetization of the
• Two media types:
g-Fe2O3. +/- mag. moment
along axis. (tape, floppy)
--Thin film: CoPtCr or CoCrTa
alloy. Domains are ~ 10-30nm!
Sheet resistivity is the resistivity divided by the thickness of the doped region, and is denoted W/□
rs is the sheet resistivity
If we know the area per square, the resistance is
Charge carriers follow a random path unless an external field is applied. Then, they acquire a drift velocity that is dependent upon their mobility,mnand the strength of the field,x
Vd = -mnx
The average drift velocity, vav is dependent
Upon the mean time between collisions, 2t
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 holesJ = q(nmn + pmp) x
OHM’s Law: V = IR
Resistance, R(W) is an extrinsic quantity. Resistivity, r(Wm), is the corresponding intrinsic property.
r = R*A/l
Conductivity, s, is the reciprocal of resistivity: s(Wm)-1 = 1/r