Solid State ElectronicsText BookBen. G. Streetman and Sanjay Banerjee: Solid StateElectronic Devices, Prentice-Hall of India PrivateLimited.

Chapter 3 and 4

Metals, Semiconductors, and InsulatorsInsulator: A very poor conductor of electricity iscalled an insulator.

In an insulator material the valance bandis filled while the conduction band is empty.

The conduction band and valance band inthe insulator are separated by a large forbiddenband or energy gap (almost 10 eV).

In an insulator material, the energy whichcan be supplied to an electron from a applied fieldis too small to carry the particle from the fieldvalance band into the empty conduction band.

Since the electron cannot acquiresufficient applied energy, conduction isimpossible in an insulator.

Semiconductor: A substance whose conductivity lies betweeninsulator and conductor is a semiconductor.

A substance for which the width of the forbidden energyregion is relatively small (almost 1 eV) is called semiconductor.In a semiconductor material, the energy which can be supplied toan electron from a applied field is too small to carry the particlefrom the field valance band into the empty conduction band at 0K.As the temperature is increased, some of the valance bandelectrons acquire thermal energy. Thus, the semiconductors allowfor excitation of electrons from the valance band to conductionband.These are now free electrons in the sense that they can moveabout under the influence of even a small-applied field.

Metal: A metal is an excellent conductor.In metals the band either overlap or are only partially

filled.Thus electrons and empty energy states are intermixed within thebands so that electrons can move freely under the influence of anelectric field.

Direct Material: The material (such asGaAs) in which a transition of an electronfrom the minimum point of conduction bandto the maximum point of valence band takesplace with the same value of K (propagationconstant or wave vector) is called directsemiconductor material.

According to Eq. (3-1) the energy (E) vspropagation constant (k) curve is shown inthe figure.

A direct semiconductor such as GaAs, anelectron in the conduction band can fall to anempty state in the valence band, giving offthe energy difference Eg as a photon of light.

Direct and Indirect Semiconductors

Indirect Material: The material (such as Si) inwhich a transition of an electron from theminimum point of conduction band to themaximum point of valence band takes placewith the different values of K (propagationconstant or wave vector) is called indirectmaterial.

According to Eq. (3-1) the energy (E) vspropagation constant (k) curve is shown in thefigure.

An electron in the conduction band minimum of an indirectsemiconductor cannot fall directly to the valence band maximum butmust undergo a momentum change as well as changing its energy.It may go through some defect state (Et) within the band gap.In an indirect transition which involves a change in k, the energy isgenerally given up as heat to the lattice rather than as emitted photon.

Intrinsic MaterialA perfect semiconductor with no impurities or lattice defect iscalled an intrinsic material.In intrinsic material, there are no charge carrier at 0K, since thevalence band is filled with electrons and the conduction band isempty.At high temperature electron-hole pairs are generated as valenceband electrons are excited thermally across the band gap to theconduction band.These EHPs are the only charge carriers in intrinsic material.Since the electrons and holes are crated in pairs, the conductionband electron concentration n (electron/cm3) is equal to theconcentration of holes in the valence band p (holes/cm3).Each of these intrinsic carrier concentrations is commonlyreferred to as ni. Thus for intrinsic material: n=p=ni (3-6)At a temperature there is a carrier concentration of EHPs ni.

Recombination is occurs when an electron in the conductionband makes transition to an empty state (hole) in the valenceband, thus annihilating the pair.If we denote the generation rate of EHPs as gi (EHP/cm3-s)and the recombination rate ri, equilibrium requires thatri=gi (3-7a)Each of these rates is temperature dependent.gi(T) increases when the temperature is raised, and a newcarrier concentration ni is established such that the higherrecombination rate ri(T) just balances generation.At any temperature, the rate of recombination of electronsand holes ri is proportional to the equilibrium concentrationof electrons n0 and the concentration of holes p0:ri=rn0p0= rni2=gi (3-7b)The factor r is a constant of proportionality which dependson the particular mechanism takes place.

Extrinsic MaterialWhen a crystal is doped such that the equilibrium carrierconcentrations n0 and p0 are different from carrier concentrationni, the material is said to be extrinsic material.In addition to the intrinsic carriers generated, it is possible tocreate carriers in semiconductors by purposely introducingimpurities into the crystal.

This process, called doping, is the most common technique forvarying conductivity of semiconductor.

There are two types of doped semiconductors, n-type (mostlyelectrons) and p-type (mostly holes).An impurity from column V of the periodic table (P, As and Sb)introduces an energy level very near the conduction band in Ge orSi.

The energy level very near the conduction band is filled with electronsat 0K, and very little thermal energy is required to excite theseelectrons to the conduction band (Fig. 3-12a).Thus at 50-100K virtually all of the electrons in the impurity level are,donated to the conduction band.

Such an impurity level is called a donor level and the column Vimpurities in Ge or Si are called donor impurities.Semiconductors doped with a significant number of donor atoms willhave n0>>(ni,p0) at room temperature.

Fig. 3-12 (a) Donation ofelectrons from donor level toconduction band.

This is n-type material.

Similarly, an impurity from column III of the periodic table (B, Al, Gaand In) introduces an energy level very near the valence band in Ge orSi.These levels are empty of electrons at 0K (Fig. 3-12b).At low temperatures, enough thermal energy is available to exciteelectrons from the valence into the impurity level, leaving behind holesin the valence band.Since this type of impurity level accepts electrons from the valenceband, it is called an acceptor level, and the column III impurities areacceptor impurities in the Ge and Si.

Fig. 3.12b

Doping with acceptorimpurities can create asemiconductor with a holeconcentration p0 much greaterthat the conduction bandelectron concentration n0.This type is p-type material.

Carrier concentrationThe calculating semiconductor properties and analyzing devicebehavior, it is often necessary to know the number of charge carriersper cm3 in the material.To obtain equation for the carrier concentration, Fermi-Diracdistribution function can be used.The distribution of electrons over a range of allowed energy levels atthermal equilibrium is

kTFEEeEf /)(1

1)(

where, k is Boltzmanns constant (k=8.2610-5 eV/K=1.3810-23J/K).The function f(E), the Fermi-Dirac distribution function, gives theprobability that an available energy state at E will be occupied by anelectron at absolute temperature T.The quantity EF is called the Fermi level, and it represents animportant quantity in the analysis of semiconductor behavior.

For an energy E equal to the Fermi level energy EF, the occupationprobability is

21

11)( /)(

kTFEFEF eEf

The significant of Fermi Level is that the probability of electron andhole is 50 percent at the Fermi energy level. And, the Fermi functionis symmetrical about EF for all temperature; that is, the probabilityf(EF +E) of electron that a state E above EF is filled is the same asprobability [1-f(EF-E)] of hole that a state E below EF is empty.

At 0K the distribution takes thesimple rectangular form shown inFig. 3-14.With T=0K in the denominator ofthe exponent, f(E) is 1/(1+0)=1when the exponent is negative(EEF).

This rectangular distribution implies that at 0K every available energystate up to EF is filled with electrons, and all states above EF are empty.At temperature higher than 0K, some probability exists for states abovethe Fermi level to be filled.At T=T1 in Fig. 3-14 there issome probability f(E) that statesabove EF are filled, and there isa corresponding probability [1-f(E)] that states below EF areempty.

The symmetry of thedistribution of empty and filledstates about EF makes the Fermilevel a natural reference pointin calculations of electron andhole concentration insemiconductors.

Fig. 3-15(a) Intrinsic Material

For intrinsic material, the concentration of holes inthe valence band is equal to the concentration ofelectrons in the conduction band.

Therefore, the Fermi level EF must lies at the middleof the band gap.

Since f(E) is symmetricalabout EF, the electronprobability tail if f(E)extending into the conductionband of Fig. 3-15a issymmetrical with the holeprobability tail [1-f(E)] in thevalence band.

In n-type material the Fermi level lies nearthe conduction band (Fig. 3-15b) such thatthe value of f(E) for each energy level in theconduction band increases as EF movescloser to Ec.Thus the energy difference (Ec- EF) givesmeasure of n.

Fig. 3.15(b) n-type material

Fig. 3.15(c) p-type material

In p-type material the Fermi level liesnear the valence band (Fig. 3-15c) suchthat the [1- f(E)] tail value Ev is largerthan the f(E) tail above Ec.The value of (EF-Ev) indicates howstrongly p-type the material is.

Example: The Fermi level in a Si sample at equilibrium is located at0.2 eV below the conduction band. At T=320K, determine theprobability of occupancy of the acceptor states if the acceptor statesrelocated at 0.03 eV above the valence band.

Solution:

From above figure, Ea-EF={0.03-(1.1-0.2)} eV= -0.87 eVkT= 8.6210-5 eV/K320=2758.4 eVwe know that,

0.11

11

1)()5104.2758/(87.0/)(

ee

Ef kTFEaEa

Electron and Hole Concentrations at EquilibriumThe concentration of electron and hole in the conduction band andvalance are

where N(E)dE is the density of states (cm-3) in the energy range dE.The subscript 0 used with the electron and hole concentration symbols(n0, p0) indicates equilibrium conditions.The number of electrons (holes) per unit volume in the energy rangedE is the product of the density of states and the probability ofoccupancy f(E) [1-f(E)].Thus the total electron (hole) concentration is the integral over theentire conduction (valance) band as in Eq. (3.12).The function N(E) is proportional to E(1/2), so the density of states inthe conduction (valance) band increases (decreases) with electron(hole) energy.

)12.3()()](1[

)12.3()()(

0

0

bdEENEfp

adEENEfnvE

cE

Similarly, the probability of finding an empty state (hole) in thevalence band [1-f(E)] decreases rapidly below Ev, and most holeoccupy states near the top of the valence band.

This effect is demonstrated for intrinsic, n-type and p-type materialsin Fig. 3-16.

Fig. 3.16 (a) Concentration of electrons and holes in intrinsic material.

Fig. 3.16 (b) Concentration of electrons and holes in n-type material.

Fig. 3.16 (a) Concentration of electrons and holes in p-type material.

The electron and hole concentrations predicted by Eqs. (3-15) and (3-18) are valid whether the material is intrinsic or doped, providedthermal equilibrium is maintained.

Thus for intrinsic material, EF lies at some intrinsic level Ei near themiddle of the band gap, and the intrinsic electron and holeconcentrations are

)21.3(, /)(/)( kTvEiEvikTiEcEci eNpeNn

From Eqs. (3.15) and (3.18), we obtain

kTvEFEvkTFEcEc eNeNpn /)(/)(00

)22.3(//)(00kTgE

vckTvEcEvc eNNeNNpn

From Eq. (21), we obtain

)23.3(//)( kTgEvckTvEcEvcii eNNeNNpn

kTvEiEvkTiEcEcii eNeNpn /)(/)(

From Eqs. (3.22) and (3.23), the product of n0 and p0 at equilibrium isa constant for a particular material and temperature, even if thedoping is varied.

The intrinsic electron and hole concentrations are equal, ni=pi; thusfrom Eq. (3.23) the intrinsic concentrations is

)24.3(2/ kTgEvci eNNn

The constant product of electron and hole concentrations in Eq. (3.24)can be written conveniently from (3.22) and (3.23) as

)25.3(200 inpn

At room temperature (300K) is: For Si approximately ni=1.51010cm-3; For Ge approximately ni=2.51013 cm-3;

)26.3(/)(

/)(

kTvEiEiv

kTiEcEic

epNenN

From Eq. (3.21), we can write as

kTFEcEiEcEikTFEcEkTiEcEi eneenn /)(/)(/)(0

Substitute the value of Nc from (3.26) into (3.15), we obtain

)27.3(/)(/)(0kTiEFEi

kTFEiEi enenn Substitute the value of Nv from (3.26) into (3.18), we obtain

kTvEFEvEiEikTvEFEkTvEiEi eneepp /)(/)(/)(0

)28.3(/)(/)(0kTFEiEi

kTiEFEi enenp It seen from the equation (3.27) that the electron concentrations n0increases exponentially as the Fermi level moves away from Eitoward the conduction band.

Similarly, the hole concentrations p0 varies from ni to larger values asEF moves from Ei toward the valence band.

Temperature Dependence of Carrier Concentrations

The variation of carrier concentration with temperature is indicatedby Eq. (3.21)

The intrinsic carrier ni has a strong temperature dependence (Eq.3.24) and that EF can vary with temperature.

)21.3(, /)(/)( kTvEiEvikTiEcEci eNpeNn

)24.3(2/ kTgEvci eNNn

The temperature dependence of electron concentration in a dopedsemiconductor can be visualized as shown in Fig. 3-18.

Figure 3-18 Carrier concentration vs.inverse temperature for Si doped with1015 donors/cm3.

In this example, Si is dopedn-type with donorconcentration Nd of 1015 cm-3.At very low temperature(large 1/T) negligible intrinsicEHPs exist, and the donorelectrons are bound to thedonor atoms.

As the temperature is raised,these electrons are donated tothe conduction band, and atabout 100K (1000/T=10) allthe donor atoms are ionized.This temperature range iscalled ionization region.Once the donor atoms are ionized, the conduction band electronconcentration is n0Nd=1015 cm-3, since one electron is obtained for eachdonor atom.

When every available extrinsic electron has been transferred to theconduction band, no is virtually constant with temperature until theconcentration of intrinsic carriers ni becomes comparable to theextrinsic concentration Nd.Finally, at higher temperature ni is much greater than Nd, and theintrinsic carriers dominate.

In most devices it is desirable tocontrol the carrier concentration bydoping rather than by thermal EHPgeneration.

Thus one usually dopes the materialsuch that the extrinsic range extendsbeyond the highest temperature atwhich the device to be used.

Excess Carrier in SemiconductorsThe carriers, which are excess of the thermal equilibrium

carries values, are created by external excitation is called excesscarriers.

The excess carriers can be created by optical excitation orelectron bombardment.

Optical AbsorptionMeasurement of band gap energy: The band gap energy of asemiconductor can be measured by the absorption of incident photonsby the material.

In order to measure the band gap energy, the photons of selectedwavelengths are directed at the sample, and relative transmission of thevarious photons is observed.

This type of band gap measurement gives an accurate value ofband gap energy because photons with energies greater than the bandgap energy are absorbed while photons with energies less than band gapare transmitted.

Excess carriers by optical excitation: Itis apparent from Fig. 4-1 that a photonwith energy hv>Eg can be absorbed in asemiconductor.

Figure 4-1 Optical absorption of a photon withhv>Eg: (a) an EHP is created during photonabsorption (b) the excited electron gives upenergy to the lattice by scattering events; (c)the electron recombines with a hole in thevalence band.

Thus the excited electron losses energy to the lattice in scattering events untilits velocity reaches the thermal equilibrium velocity of other conduction bandelectrons.

The electron and hole created by this absorption process are excess carriers:since they are out of balance with their environment, they must even eventuallyrecombine.

While the excess carriers exit in their respective bands, however, they are freeto contribute to the conduction of material.

Since the valence band containsmany electrons and conduction band hasmany empty states into which theelectron may be excited, the probabilityof photon absorption is high.

Fig. 4-1 indicates, an electronexcited to the conduction band by opticalabsorption may initially have moreenergy than is common for conductionband electrons.

If a beam of photons with hv>Eg falls on asemiconductor, there will be some predictable amount ofabsorption, determined by the properties of the material.

The ratio of transmitted to incident lightintensity depends on the photon wavelength and thethickness of the sample.

let us assume that a photon beam of intensity I0 (photons/cm-2-s) is directedat a sample of thickness l as shown in Fig. 4-2.

)1.4()()( xdxxd II

The beam contains only photons of wavelength selected bymonochromator.

As the beam passes through the sample, its intensity at a distance x fromthe surface can be calculated by considering the probability of absorption with inany increment dx.The degradation of the intensity dI(x)/dx is proportional to the intensity remainingat x:

I0 It

The solution to this equation is)2.4(0)(

xex II

Figure 4-3 Dependence of opticalabsorption coefficient for asemiconductor on the wavelengthof incident light.

and the intensity of light transmittedthrough the sample thickness l is

)3.4(0let II

The coefficient is called theabsorption coefficient and has units ofcm-1.

This coefficient varies with the photonwavelength and with the material.

Fig. 4-3 shows the plot of vs. wavelength. There is negligible absorption at long wavelength (hv small) andconsiderable absorptions with energies larger than Eg.The relation between photon energy and wavelength is E=hc/. If E isgiven in electron volt and is micrometers, this becomes E=1.24/.

Steady State Carrier GenerationThe thermal generation of EHPs is balanced by the recombination rate that means[Eq. 3.7] )10.4(00

2)( pnrinrTg If a steady state light is shone on the sample, an optical generation rate gop will beadded to the thermal generation, and the carrier concentration n and p will increase tonew steady sate values.

)11.4()0)(0()( ppnnrnpropgTg For steady state recombination and no traping, n=p; thus Eq. (4.11) becomes

)12.4(]2)00[(00)( nnpnrpnropgTg Since g(T)==rn0p0 and neglecting the n2, we can rewrite Eq. (4.12) as

)13.4()/(])00[( nnnpnropg

.theis)00(

1where, timelifecarrierpnrn

If n and p are the carrier concentrations which are departed from equilibrium:

The excess carrier can be written as )14.4(nopgpn

Quasi-Fermi Level

The Fermi level EF used in previous equations is meaningful only when no excesscarriers are present.

The steady state concentrations in the same form as the equilibrium expressions bydefining separate quasi-Fermi levels Fn and Fp for electrons and holes.The resulting carrier concentration equations

)15.4(/)(

;/)(KTpFiEeinpKTiEnFeinn

can be considered as defining relation for the quasi-Fermi levels.