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### 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

ELECTRICAL CONDUCTION

• Ohm's Law:

DV = I R

voltage drop (volts)

resistance (Ohms)

current (amps)

• Resistivity, r and Conductivity, s:

--geometry-independent forms of Ohm's Law

E: electric

field

intensity

resistivity

(Ohm-m)

J: current density

conductivity

• Resistance:

3

Resistivity and Conductivity as charged particles

mobility, m =

Where

is the average velocity

is the average distance between collisions,

divided by the average time between collisions,

t

d

CONDUCTION IN TERMS OF ELECTRON AND HOLE MIGRATION

• Concept of electrons and holes:

• Electrical Conductivity given by:

hole mobility

# electrons/m3

# holes/m3

electron mobility

11

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.

CONDUCTION & ELECTRON TRANSPORT

• 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.

6

ENERGY STATES: INSULATORS AND SEMICONDUCTORS

• Semiconductors:

--Higher energy states

separated by a smaller gap.

• Insulators:

--Higher energy states not

accessible due to gap.

7

PURE SEMICONDUCTORS: CONDUCTIVITY VS T

• Data for Pure Silicon:

--s increases with T

--opposite to metals

electrons

can cross

gap at

higher T

material

Si

Ge

GaP

CdS

band gap (eV)

1.11

0.67

2.25

2.40

10

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

• 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)

12

Equations describing Intrinsic and Extrinsic conduction

- Using the Fermi-Dirac equation, we can find the number of charge carrier per unit volume as:
- Ne = Noexp(-Eg/2kT)
- Nois a preexponential function,
- Egis the band-gap energy and
- kis 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:
- s(T) = so 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

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 conduction...

--extrinsic doping level:

1021/m3 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"

13

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.

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:

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, 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:

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.

• 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.

14

Piezoelectrics

Field produced by stress:

x = electric field

s= applied stress

E=Elastic modulus

d = piezoelectric constant

g = constant

Strain produced by field:

Elastic modulus:

APPLIED MAGNETIC FIELD

• Created by current through a coil:

• Relation for the applied magnetic field, H:

current

applied magnetic field

units = (ampere-turns/m)

2

MAGNETIC SUSCEPTIBILITY

• Measures the response of electrons to a magnetic

field.

• Electrons produce magnetic moments:

Adapted from Fig. 20.4, Callister 6e.

• Net magnetic moment:

--sum of moments from all electrons.

• Three types of response...

4

MAGNETIC STORAGE

recording medium

recording head

• 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

medium.

• Two media types:

--Particulate: needle-shaped

g-Fe2O3. +/- mag. moment

along axis. (tape, floppy)

--Thin film: CoPtCr or CoCrTa

alloy. Domains are ~ 10-30nm!

(hard drive)

9

Sheet Resistivity

R =

=

=

=

=

Sheet resistivity is the resistivity divided by the thickness of the doped region, and is denoted W/□

rs is the sheet resistivity

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,mnand the strength of the field,x

Vd = -mnx

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 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

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