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

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• Ohm's Law:V = I R

voltage drop (volts) resistance (Ohms)current (amps)

VIe-

A(cross sect. area)

L

• Resistivity, and Conductivity, : --geometry-independent forms of Ohm's Law

VL

IAE: electric

fieldintensity

resistivity(Ohm-m)

J: current density

I

conductivity• Resistance: R

LA

L

A

ELECTRICAL CONDUCTION

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Resistivity and Conductivity as charged particles

mobility, =E

V

V

Where is the average velocity

is the average distance between collisions,

divided by the average time between collisions,

V

d

t

t

dV t

d

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• Electrical Conductivity given by:

nee peh

11

# electrons/m3

electron mobility# holes/m3

hole mobility

• Concept of electrons and holes:

+ -

electron hole pair creation

+ -

no applied electric field

applied electric field

valence electron Si atom

applied electric field

electron hole pair migration

CONDUCTION IN TERMS OF ELECTRON AND HOLE MIGRATION

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Polystyrene <10-14

Polyethylene 10-15-10-17

Silver 6.8 x 107

Copper 6.0 x 107

Iron 1.0 x 107

METALS

Silicon 4 x 10-4

Germanium 2 x 100

GaAs 10-6

SEMICONDUCTORS

Soda-lime glass 10-10

Concrete 10-9

Aluminum oxide <10-13

CERAMICS

POLYMERS

conductors

semiconductors insulators

4

• Room T values (Ohm-m)-1CONDUCTIVITYCOMPARISON

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.

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• Metals:-- Thermal energy puts many electrons into a higher energy state.

+-

net e- flow

• Energy States:-- the cases below for metals show that nearby energy states are accessible by thermal fluctuations.

Energy

filled band

filled valence band

empty band

filled s

tate

s

CONDUCTION & ELECTRON TRANSPORT

filled band

Energy

partly filled valence band

empty band

GAP

filled s

tate

s

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• Insulators: --Higher energy states not accessible due to gap.

• Semiconductors: --Higher energy states separated by a smaller gap.

Energy

filled band

filled valence band

empty band

filled s

tate

s

GAP

Energy

filled band

filled valence band

empty band

filled s

tate

s

GAP?

ENERGY STATES: INSULATORS AND SEMICONDUCTORS

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• Data for Pure Silicon: -- increases with T --opposite to metals

kTEundoped

gape 2/Energy

filled band

filled valence band

empty band

fille

d s

tate

s

GAP?

electronscan crossgap athigher T

materialSiGeGaPCdS

band gap (eV)1.110.672.252.40

PURE SEMICONDUCTORS: CONDUCTIVITY VS T

electrical conductivity,

(Ohm-m)-1

50 100 100010-210-1

100

101

102

103104

pure (undoped)

T(K)

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)

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• 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)

nee peh

no applied electric field

5+

4+ 4+ 4+ 4+

4+

4+4+4+4+

4+ 4+

Phosphorus atom

no applied electric field

Boron atom

valence electron

Si atom

conduction electron

hole

3+

4+ 4+ 4+ 4+

4+

4+4+4+4+

4+ 4+

INTRINSIC VS EXTRINSIC CONDUCTION

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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)

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 insulatorIf 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

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doped 0.0013at%B

0.0052at%B

ele

ctri

cal co

nd

uct

ivit

y,

(Oh

m-m

)-1

50 100 100010-210-1

100

101

102

103104

pure (undoped)

T(K)

13

• 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: 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"

con

du

ctio

n e

lect

ron

co

nce

ntr

ati

on

(1

021 /

m3)

T(K)60040020000

1

2

3

freeze

-ou

t

extr

insi

c

intr

insi

c

dopedundoped

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/ = qdNd

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

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 MassIn general, the curve of Energy vs. k is non-

linear, with E increasing as k increases.

E = ½ mv2 = ½ p2/m = h2/4m k2

We can see that energy varies inversely with mass. Differentiating E wrt k twice, and solving for mass gives:

2

2

2*

2dkEd

hm

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:

tln mmm

21

3

11*

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• 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-type n-type

++

++

+

---

--

p-type n-type+ -

+++

+

+

---

--

p-type n-type- +

P-N RECTIFYING JUNCTION

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Piezoelectrics

Field produced by stress:

gStrain produced by field:

d

Elastic modulus: gd

E1

= electric field

= applied stress

E=Elastic modulus

d = piezoelectric constant

g = constant

2

• Created by current through a coil:

Applied magnetic field H

current I

N turns total

L = length of each turn

• Relation for the applied magnetic field, H:

H

NIL

applied magnetic fieldunits = (ampere-turns/m)

current

APPLIED MAGNETIC FIELD

HB oo

)1( moo HB

4

• Measures the response of electrons to a magnetic field.• Electrons produce magnetic moments:

magnetic moments

electron

nucleus

electron

spin

• 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 -Fe2O3. +/- mag. moment along axis. (tape, floppy)

~2.5m

--Thin film: CoPtCr or CoCrTa alloy. Domains are ~ 10-30nm! (hard drive)

~60nm

MAGNETIC STORAGE

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Sheet Resistivity

R =

I

V

A

Lw

L

t

JA

EL= =

w

Ls= =

s is the sheet resistivitySheet 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 squareareasquaresns /

ConductivityCharge 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,

Vd = -n

The average drift velocity, vav is dependentUpon 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 = -nqvd = nqn

n is the number of charges, and

q is the charge (1.6 x 10-19

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(nn + pp)