1 Energy Bands & Charge Carriers in Semiconductors

8/11/2019 1 Energy Bands & Charge Carriers in Semiconductors

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Energy Bands & Charge Carriers

in Semiconductors


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Atomic Electron Energy Levels


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Bonding Forces in Solids

The interaction of electrons in neighboring atoms of

a solid serves the very important function of holding

the crystal together. For example, alkali halides such

as NaCl are typified by ionic bonding.

In the NaCl lattice, each Na atom is surrounded by six

nearest neighbor CI atoms, and vice versa. Four of

the nearest neighbors are evident in the two-

dimensional representation.


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An important observation in the NaCl structure is

that all electrons are tightly bound to atoms. Once

the electron exchanges have been made between

the Na and CI atoms to form the Na+ and CI" ions,the outer orbits of all atoms are completely filled.

There are no loosely bound electrons to participate

in current flow; as a result, NaCl is a good insulator.


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In a metal atom the outer electronic shell is only

partially filled, usually by no more than three

electrons. This electron is loosely bound and is given

up easily in ion formation.

The forces holding the lattice together arise from an

interaction between the positive ion cores and the

surrounding free electrons. This is one type of

metallic bonding.


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A third type of bonding is exhibited by the diamond

lattice semiconductors. The bonding forces arise

from a quantum mechanical interaction between the

shared electrons. This is known as covalent bonding;each electron pair constitutes a covalent bonding.


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Energy Bands

The forces of attraction and repulsion between

atoms will find a balance at the proper inter atomic

spacing for the crystal.

In the process, important changes occur in theelectron energy level configurations, and these

changes result in the varied electrical properties of

solids.


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In a solid, many atoms are brought together, so that

the split energy levels form essentially continuous

bands of energies.


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Energy levels are separated by an energy gap Eg

wide, which contains no allowed energy levels for

electrons to occupy. This gap is sometimes called a"forbidden band," .


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At 0 K the electrons will occupy the lowest energy

states available to them.

Thus at 0 K, every state in the valence band will be

filled, while the conduction band will be completelyempty of electrons.


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Metals, Semiconductors, and

Insulators

For electrons to experience acceleration in an

applied electric field, they must be able to move into

new energy states. This implies there must be empty

states (allowed energy states which are not alreadyoccupied by electrons) available to the electrons.

The silicon band structure is such that the valence

band is completely filled with electrons at 0 K and

the conduction band is empty.


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There are no electrons in the conduction band, so nocharge transport can take place there either. Thussilicon has a high resistivity typical of insulators.

Semiconductor materials at 0 K have basically thesame structure as Insulators - a filled valence bandseparated from an empty conduction band by a band

gap containing no allowed energy states.

The difference lies in the size of the band gap Eg,


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Valence Band : the highest range of electron energieswhere electrons are normally present at absolute zero.

: this is the highest filled band

Conduction Band : the range of electron energy sufficient to

make the electrons free to accelerate under the influence of

an applied electric field (i.e., current).

: this is the lowest unfilled band


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Band Gap Comparisons- the following shows the relationship of Band Gap energies between insulators,

semiconductors, and metals

- notice that the only difference between an insulator and a semiconductor is

that the band gap is smaller in a semiconductor.

- notice that there is an overlap between the conduction and valence bands in

metals. This means that metals are always capable of conducting current.


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Band Gap Comparisons

Insulator Band Gap : it is large enough so that at ordinarytemperatures, no electrons reach the conduction band.

Semiconductor Band Gap: it is small enough so that at ordinary

temperatures, thermal energy can give an electron enough energy to

jump to the conduction band ,we can also change the semiconductorinto a conductor by introducing impurities


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For example, the semiconductor Si has a band gap of

about 1.1 eV compared with 5 eV for diamond.

Thus an important difference between

semiconductors and insulators is that the number ofelectrons available for conduction can be increased

greatly in semiconductors by thermal or optical

energy.

In metals the bands either overlap or are only

partially filled.


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Direct Semiconductor

In a direct semiconductor such as GaAs, anelectron in the conduction band can fall to an

empty state in the valence band, giving off the

energy difference Eg as a photon of light.


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Indirect Semiconductor

On the other hand, in indirect semiconductor such as

Si electron in the conduction band cannot fall directly

to the valence band maximum but must undergo a

momentum change as well as changing its energy.

In an indirect transition which involves a change in

the energy, is generally given up as heat rather than

as an emitted photon.


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With N electrons/cm3 in the band , we express the current density using a sum of

all of the electron velocities, and including the charge q on each electron.

In a unit volume,


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An empty conduction band completely devoid of electrons or

a valence band completely full of electrons cannot give rise to

a net motion of electrons, and thus to current conduction.

Similarly, a few electrons in an otherwise empty conduction

band move opposite to an electric field, while holes in an

otherwise filled valence band move in the direction of the

field.

We can account for the current flow in a semiconductor by

the motion of these two types of charge carriers.


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Effective Mass

The electrons in a crystal are not completely free, butinstead interact with the periodic potential of thelattice.

As a result, their motion cannot be expected to bethe same as for electrons in free space.

Thus, in applying the usual equations ofelectrodynamics to charge carriers in a solid, wemust use altered values of particle mass.


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In doing so. we account for most of the influences of

the lattice, so that the electrons and holes can be

treated as "almost free" carriers in most

computations.

The calculation of effective mass must take into

account the shape of the energy bands in three-

dimensional space.


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Fermi Level

- the Fermi Level (or energy) represents an energy level that atabsolute zero:

- all bands below this level are filled- all bands above this level are unfilled

- the Fermi Level at room temperatures is the energy at which theprobability of a state being occupied has fallen to 0.5

- at higher temperatures, in order for an electron to be used ascurrent, it needs to have an energy level close to the Fermi Level

- this can also be thought of as the equilibrium point of the material


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Band Diagram - in a band diagram, we tabulate the relative locations of important energy

levels

- Note that EOis where the electron has enough energy to leave thematerial all together (an example would be a CRT monitor)

- as electrons get enough energy to reach near the Fermi level, conduction

begins to occur


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Intrinsic Material

A perfect semiconductor crystal with no impurities orlattice defects is called an intrinsic semiconductor.

In such material there are no charge carriers at 0 K,since the valence band is filled with electrons and the

conduction band is empty.

At higher temperatures electron-hole pairs aregenerated as valence band electrons are excitedthermally across the band gap to the conduction band.

These EHPs are the only charge carriers in intrinsicmaterial.


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The generation of EHPs can be visualized in aqualitative way by considering the breaking ofcovalent bonds in the crystal lattice .

If one of the Si valence electrons is brokenaway from its position in the bondingstructure such that it becomes free to moveabout in the lattice, a conduction electron is

created and a broken bond (hole) is leftbehind


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The energy required to break the bond is the

band gap energy Eg.


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Band Diagram of Intrinsic Silicon

- Intrinsic Silicon has a band gap energy of 1.1 eV

- @ 0 K, Eg=1.17 eV

- @ 300 K, Eg=1.14 eV


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At a given temperature there is a certain

concentration of electron - hole pairs ni

Obviously, if a steady state carrier

concentration is maintained, there must berecombination of EHPs at the same rate at

which they are generated.


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If we denote the generation rate of EHPs as gi(EHP/cm3-s) and the recombination rate as ri

,equilibrium requires that

ri= gi

For example ,gi increases when thetemperature is raised, and a new carrierconcentration ni, is established such that thehigher recombination rate ri(T) just balancesgeneration.


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At any temperature, we can predict that the

rate of recombination of electrons and holes ri

is proportional to the equilibrium

concentration of electrons n0 and theconcentration of holesp0:

The factor is a constant of proportionality.


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Extrinsic Material


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Extrinsic Material

In addition to the intrinsic carriers generated

thermally, it is possible to create carriers in

semiconductors by purposely introducing

impurities into the crystal.

This process, called doping.


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Thus there are two types of doped

semiconductors, n-type (mostly electrons) and

p-type (mostly holes).

For example, an impurity from column V of

the periodic table (P, As, and Sb) introduces an

energy level very near the conduction band in

Ge or Si.


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This level is filled with electrons at 0 K, andvery little thermal energy is required to excitethese electrons to the conduction band .

Thus at about 50-100 K virtually all of theelectrons in the impurity level are "donated"to the conduction band.

Such an impurity level is called a donor level,and the column V impurities in Ge or Si arecalled donor impurities.


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Thus semiconductors doped with a significant

number of donor atoms will have n0 (ni,p0)

at room temperature.

This is n-type material.


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Atoms from column III (B, Al, Ga, and In)

introduce impurity levels in Ge or Si near the

valence band.

At low temperatures, enough thermal energy

is available to excite electrons from the

valence band into the impurity level, leaving

behind holes in the valence band.


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Generally, the column V donor levels lie

approximately 0.01 eV below the conduction

band in Ge, and the column III acceptor levels

lie about 0.01 eV above the valence band. In Sithe usual donor and acceptor levels lie about

0.03-0.06 eV from a band edge.


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ni= 1.45 x 1010/cm3


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We can approximate energy required to excite the fifth

electron of a donor( donor binding energy ) atom into theconduction band by the above equation.


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ni= 1.45 x 1010

/cm3


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The Fermi Level

The distribution of electrons over a range of allowedenergy levels at thermal equilibrium is

where k is Boltzmann's constant (k = 8.62 x 10~5 eV/K =1.38 X 10~23 J/K).


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The function f(E), the Fermi-Dirac distribution

function, gives the probability that an

available energy state at E will be occupied by

an electron at absolute temperature T.

The quantity EFis called the Fermi level, and it

represents an important quantity in theanalysis of semiconductor behavior.


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We notice that, for an energy E equal to the Fermi levelenergy EF, the occupation probability is

Thus an energy state at the Fermi level has aprobability of 1/2 of being occupied by an electron.


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A closer examination of f(E) indicates that at 0K the distribution takes the simple rectangularform shown in Fig.

With T = 0 in the denominator of theexponent, f(E) is 1/(1 + 0) = 1 when theexponent is negative (E < EF), and

when the exponent is positive (E > EF).


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This rectangular distribution implies that at 0 K every

available energy state up to EFis filled with electrons,

and all states above EFare empty.


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In applying the Fermi-Dirac distribution to

semiconductors, we must recall that f(E) is the

probability of occupancy of an available state

at E. Thus if there is no available state at E (e.g., in

the band gap of a semiconductor), there is no

possibility of finding an electron there.


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The tails in f(E) are exaggerated in Fig. forillustrative purposes. Actually, the probabilityvalues at Ev and Ec are quite small for intrinsic

material at reasonable temperatures.

Probability of occupancy f(E) for an individualstate in the conduction band and the holeprobability [1 f(E)] for a state in the valenceband are quite small.


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Because of the relatively large density of

states in each band, small changes in f(E) can

result in significant changes in carrier

concentration.


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For p-type material the Fermi level lies near

the valence band such that the [1f(E)] tail

below Evis larger than the f(E) tail above Ec.

The value of (EFEv) indicates how strongly

p-type the material is.

Electron and Hole Concentrations at


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Electron and Hole Concentrations at

Equilibrium

The Fermi distribution function can be used to calculatethe concentrations of electrons and holes in asemiconductor, if the densities of available states in thevalence and conduction bands are known

where N(E)dE is the density of states in the energy rangedE


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The number of electrons per unit volume in

the energy range dE is the product of the

density of states and the probability of

occupancyf(E).

Thus the total electron concentration is the

integral over the entire conduction band.


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The Fermi function becomes extremely small forlarge energies.

The result is that the product f(E)N(E) decreases

rapidly above Ec, and very few electrons occupyenergy states far above the conduction bandedge.


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Similarly, the probability of finding an empty

state (hole) in the valence band [1 - f(E)]

decreases rapidly below Ev, and most holes

occupy states near the top of the valenceband


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The result of the integration is the same as thatobtained if we represent all of the distributed

electron states in the conduction band by an

effective density of states Nc located at the

conduction band edge Ec.

Therefore, the conduction band electron

concentration is simply the effective density of statesat Ec times the probability of occupancy at Ec.

n0= Ncf(Ec)


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In this expression we assume the Fermi level Ef lies at leastseveral kT below the conduction band. Then the exponential

term is large compared with unity, and the Fermi function f(Ec)

can be simplified as

Since kT at room temperature is only 0.026 eV, this is

generally a good approximation. For this condition the

concentration of electrons in the conduction band is


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The effective density of states Nc


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Schematic band diagram, density of states, Fermi-Dirac distribution, and the carrier

concentrations for

(a) intrinsic, (b) n-type, and (c) p-type semiconductors at thermal equilibrium.


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The concentration of holes in the valence band is

where Nv is the effective density of states in the valence band.

The probability of finding an empty state at Ev is

for EFlarger than Ev by several kT. From these equations, the

concentration of holes in the valence band is .


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The effective density of states in the valenceband reduced to the band edge is


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Thusfor intrinsic material, EF lies at some intrinsic level Ei near themiddle of the band gap and the intrinsic electron and hole

concentrations are

The product of n0andp0at equilibrium is a constant for a particular

material and temperature, even if the doping is varied:


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The intrinsic electron and hole concentrations are

equal (since the carriers are created in pairs), ni =pi

thus the intrinsic concentration is

The constant product of electron and hole concentrations

can be written conveniently as


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Another convenient way of writing no and po is


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A Si sample is doped with 1017 As atoms/Cm3.What is the equilibrium hole concentration p0

at 300 K. Where is EFrelative to Ei.

Assume ni= 1.5 x 1010


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T t d d f i C t ti


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Temperature dependence of carrier Concentration

The variation of carrier concentration withtemperature is indicated by the equation. Initially,the variation of n0 and p0 with T seems relativelystraightforward in these relations.

The problem is complicated, however, by the factthat n( has a strong temperature dependence andthat EFcan also vary with temperature.


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Conductivity and Mobility


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Conductivity and Mobility

The charge carriers in a solid are in constantmotion, even at thermal equilibrium.

At room temperature, for example, the

thermal motion of an individual electron may

be visualized as random scattering from lattice

vibrations, impurities, other electrons, anddefects.


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The probability of the electron in returning toits starting point after some time t is negligibly

small.

However, if a large number of electrons isconsidered there will be no preferred

direction of motion for the group of electrons

and no net current flow.


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If an electric field x is applied in the x-direction, each electron experiences a netforce -qxfrom the field.

This force may be insufficient to alterappreciably the random path of an individualelectron; the effect when averaged over all

the electrons, however, is a net motion of thegroup in the -x-direction.


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If x is the x-component of the totalmomentum of the group, the force of the field

on the n electrons/cm3 is


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The current density resulting from this net drift is justthe number of electrons crossing a unit area per unittime n(vx) multiplied by the charge on the electron (-q):

Jx= -qnvx ampere /cm2

But Jx= xThe conductivity can be written

= qnn The quantity

ncalled the electron mobility, describes

the ease with which electrons drift in the material


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The mobility can be expressed as the averageparticle drift velocity per unit electric field.

The current density can be written in terms of

mobility as


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If both electrons and holes participate, wemodify the equation as


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A given concentration of excess EHPs causes a largeshift in the minority carrier quasi-Fermi level comparedwith that for the majority carriers.

The separation of the quasi- Fermi levels Fn - Fp is adirect measure of the deviation from equilibrium.

(At equilibrium Fn= Fp= EF)

The concept of quasi-Fermi levels is very useful invisualizing minority and majority carrier concentrationsin devices.

Drift and Resistance


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Drift and Resistance

If the semiconductor bar contains both typesof carrier, the conductivity.

The resistance of the bar is then


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The physical mechanism of carrier drift requiresthat the holes in the bar move as a group in thedirection of the electric field and that theelectrons move as a group in the opposite

direction.

Both the electron and the hole components ofcurrent are in the direction of the field, since

conventional current is positive in the direction ofhole flow and opposite to the direction ofelectron flow.

Effects of Temperature and Doping on


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Mobility

The two basic types of scattering mechanismsthat influence electron and hole mobility are

lattice scattering and impurity scattering.

In lattice scattering a carrier moving through

the crystal is scattered by a vibration of the

lattice, resulting from the temperature.


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A slowly moving carrier is likely to be scatteredmore strongly by an interaction with a charged

ion than is a carrier with greater momentum.

Impurity scattering events cause a decrease in

mobility with decreasing temperature

Steady State Carrier Generation;


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Quasi-Fermi Levels

In the previous discussion we emphasized thetransient decay of an excess EHP population.

However, the various recombination

mechanisms are also important in a sample at

thermal equilibrium or with a steady state EHP

generation-recombination balance.


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A semiconductor experiences thermalgeneration of EHPs at a rate g(T) = gi

This generation is balanced by the

recombination rate so that the equilibriumconcentrations of carriers n0 and p0 are

maintained.

g(T)= rni2 =rnopo


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This equilibrium rate balance can includegeneration from defect centers as well as

band-to-band generation.

If a steady light is shone on the sample, anoptical generation rate gopwill be added to the

thermal generation, and the carrier

concentrations n and p will increase to newsteady state values.


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We can write the balance between generationand recombination in terms of the equilibrium

carrier concentrations and the departures

from equilibrium nand p. g(T) + gop= rnp =r (no + n)(po + p)


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For steady state recombination and notrapping, becomes n= p.

g(T) + gop= rnopo + r [(no + po)n) +n2]


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The term rnopois just equal to the thermalgeneration rate g(T).Thus, neglecting the n2

term for low-level excitation, we can rewrite

gop= r [(no + po)n] = n/ n


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The excess carrier concentration can bewritten as

n= p= gopn


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The Fermi level EF used is meaningful onlywhen no excess carriers are present.

However, we can write expressions for thesteady state concentrations in the same form

as the equilibrium expressions by defining

separate quasi-Fermi levels Fn and Fp forelectrons and holes.


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The resulting carrier concentration equationscan be considered as defining relations for the

quasi-Fermi levels

= ( )/

p = ()/


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In summary, the quasi-Fermi levels Fn and Fpare the steady state analogues of the

equilibrium Fermi level EF.

When excess carriers are present, the

deviations of Fn and Fp from EF indicate how

far the electron and hole populations are fromthe equilibrium values n0andpo.


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A given concentration of excess EHPs causes a largeshift in the minority carrier quasi-Fermi level comparedwith that for the majority carriers.

The separation of the quasi- Fermi levels Fn - Fp is adirect measure of the deviation from equilibrium.

(At equilibrium Fn= Fp= EF)

The concept of quasi-Fermi levels is very useful invisualizing minority and majority carrier concentrationsin devices.


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Approximate temperature dependence of mobility with both lattice and impurity

scattering


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As the concentration of impurities increases,the effects of impurity scattering are felt at

higher temperatures.


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A Si bar 0.1 cm long and 100 m2 in cross-

sectional area is doped with 1017 cm-3phosphorus. Find the current at 300 K with

10 V applied. Assume n = 700 cm2 v/s.


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n = 700 cm2 v/s

= qnno = 1.6x 10-19x 700 x1017= 11.2(.cm)-1

= 1/ = 0.0893 -cm

R = LIA = 0.0893 X 0.1/ 10-6

= 8.93 x 103

I = V/R = 10/(8.93 X 103) = 1.12 mA


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Variation of mobility with total doping impurity concentration for Ge, Si, and GaAs at 300 K.

High-Field Effects


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g

One assumption was that Ohm's law is valid in

the carrier drift processes.

That is, it was assumed that the drift current is

proportional to the electric field and that the

proportionality constant is not a function offield .


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This assumption is valid over a wide range of .

However, large electric fields (> 103 /cm) cancause the drift velocity and therefore the current

J = -qnvd to exhibit a sublinear dependence onthe electric field.

In many cases an upper limit is reached for thecarrier drift velocity in a high field.


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Saturation of electron drift velocity at high electric fields for Si

DIFFUSION OF CARRIERS


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When excess carriers are created non uniformly in asemiconductor, the electron and hole concentrationsvary with position in the sample.

Any such spatial variation (gradient) in n andp calls fora net motion of the carriers from regions of high carrierconcentration to regions of low carrier concentration.

This type of motion is called diffusion and representsan important charge transport process insemiconductors.


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When a bottle of perfume is opened in onecorner of a closed room, the scent is soondetected throughout the room.

If there is no convection or other net motionof air, the scent spreads by diffusion.

The diffusion is the natural result of therandom motion of the individual molecules.


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All the molecules undergo random thermalmotion and collisions with other molecules.

Thus each molecule moves in an arbitrarydirection until it collides with another air

molecule, after which it moves in a new

direction.


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Carriers in a semiconductor diffuse in a carrier gradientby random thermal motion and scattering from thelattice and impurities.

For example, a pulse of excess electrons injected atx =0 at time t = 0 will spread out in time as shown in Fig.

Initially, the excess electrons are concentrated atx = 0;

as time passes, however, electrons diffuse to regions oflow electron concentration until finally n(x) is constant.


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Spreading of a pulse of electrons by diffusion.


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Dnand Dp are called Electron diffusion coefficient and

hole diffusion coefficient with units Cm2/S.

The minus sign in Eq. indicates that the net motion of

electrons due to diffusion is in the direction of

decreasing electron concentration.

This is the result we expect, since net diffusion occurs

from regions of high particle concentration to regions of

low particle concentration


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The diffusion current crossing a unit area (the current

density) is the particle flux density multiplied by thecharge of the carrier.

It is important to note that electrons and holes move

together in a carrier gradient but the resulting currents

are in opposite directions because of the opposite

charge of electrons and holes.


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If an electric field is present in addition to thecarrier gradient, the current densities will

each have a drift component and a diffusion

component.


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The total current density is the sum of thecontributions due to electrons and holes.


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We can best visualize the relation betweenthe particle flow and the current by

considering a diagram such as shown in Fig.

In this figure an electric field is assumed to be

in the x-direction, along with carrier

distributions n(x) and p(x) which decrease withincreasing x.


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The resulting electron and hole diffusion currents[Jn (diff.) and Jp (diff.)] are in opposite directions.

Holes drift in the direction of the electric fieldwhereas electrons drift in the opposite directionbecause of their negative charge.

The resulting drift current is in the +x-direction ineach case.


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An important fact is that minority carriers cancontribute significantly to the current throughdiffusion.

Since the drift terms are proportional to carrierconcentration, minority carriers seldom providemuch drift current.

On the other hand, diffusion current isproportional to the gradient of concentration.


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In discussing the motion of carriers in anelectric field, we should indicate the influenceof the field on the energies of electrons in theband diagrams.

Assuming an electric field x in the x-direction,we can draw the energy bands as in Fig to

include the change in potential energy ofelectrons in the field.


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Since electrons drift in a direction opposite to the field,

we expect the potential energy for electrons to increase

in the direction of the field.


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From the definition of electric field,

we can relate x to the electron potential

energy in the band diagram by choosing some

reference in the band for the electrostatic

potential.


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Choosing Eias a convenient reference, we canrelate the electric field to this reference by


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This important equation is called the Einsteinrelation. It allows us to calculate either D or.

from a measurement of the other.

Diffusion and Recombination; The

Continuity Equation


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Continuity Equation

In the discussion of diffusion of excesscarriers, we have thus far neglected the

important effects of recombination.

These effects must be included in a

description of conduction processes, however,

since recombination can cause a variation inthe carrier distribution.


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For example, consider a differential length xof a semiconductor sample with area A in theyz-plane .

The hole current density leaving the volume,Jp(x + x), can be larger or smaller than thecurrent density entering, Jp(x), depending on

the generation and recombination of carrierstaking place within the volume.


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The net increase in hole concentration perunit time, dp/dt, is the difference between the

hole flux per unit volume entering and

leaving, minus the recombination rate.

We can convert hole current density to hole

particle flux density by dividingJpby q.

The Hall Effect


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If a magnetic field is applied perpendicular tothe direction in which holes drift in a p-typebar, the path of the holes tends to bedeflected (Fig). Using vector notation, the

total force on a single hole due to the electricand magnetic fields.

In the y direction the force is


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The Hall effect


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Therefore, to maintain a steady state flow ofholes down the length of the bar, the electric

field ymust just balance the product

so that the net force Fyis zero.


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Physically, this electric field is set up when themagnetic field shifts the hole distribution slightlyin the -y-direction.

Once the electric field ybecomes as large as et

no net lateral force is experienced by the holes asthey drift along the bar.

The establishment of the electric field y is knownas the Hall effect, and the resulting voltage

VAB=y w is called the Hall voltage.


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Thus the Hall field is proportional to the product of the

current density and the magnetic flux density.

The proportionality constant RH= 1/(qpo) is called theHall coefficient

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1 Energy Bands & Charge Carriers in Semiconductors

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