Tài liệu Semiconductor MaterialsS. K. Tewksbury Microelectronic Systems Research Center Dept. of Electrical ppt

For appropriateatomic elements, the crystalline structure leads to a disallowed energy band between theenergy level of electrons bound to the crystal’s atoms and the energy level of elec

Semiconductor Materials

S K Tewksbury Microelectronic Systems Research Center Dept of Electrical and Computer Engineering

West Virginia University Morgantown, WV 26506 (304)293-6371 Sept 21, 1995

Contents

2.1 Basic Semiconductor Materials Groups . 3

2.1.1 Elemental (IV-IV) Semiconductors . 3

2.1.2 Compound III-V Semiconductors . 4

2.1.3 Compound II-VI Semiconductors . 6

2.2 Three-Dimensional Crystal Lattice . 6

2.3 Crystal Directions and Planes . 7

3 Energy Bands and Related Semiconductor Parameters 8 3.1 Conduction and Valence Band . 9

3.2 Direct Gap and Indirect Gap Semiconductors . 12

3.3 Effective Masses of Carriers . 13

3.4 Intrinsic Carrier Densities . 14

3.5 Substitutional Dopants . 16

4 Carrier Transport 18 4.1 Low Field Mobilities . 19

4.2 Saturated Carrier Velocities . 21

5 Crystalline Defects 23 5.1 Point Defects . 23

5.2 Line Defects . 24

5.3 Stacking Faults and Grain Boundaries . 26

5.4 Unintentional Impurities . 26

5.5 Surface Defects: The Reconstructed Surface . 27

a periodic array of semiconductor atoms, i.e., within a crystalline structure For appropriateatomic elements, the crystalline structure leads to a disallowed energy band between theenergy level of electrons bound to the crystal’s atoms and the energy level of electronsfree to move within the crystalline structure (i.e., not bound to an atom) This “energygap” fundamentally impacts the mechanisms through which electrons associated with thecrystal’s atoms can become free and serve as conduction electrons The resistivity of asemiconductor is proportional to the free carrier density, and that density can be changedover a wide range by replacing a very small portion (about 1 in 106) of the base crystal’satoms with different atomic species (doping atoms) The majority carrier density is largelypinned to the net dopant impurity density By selectively changing the crystalline atomswithin small regions of the crystal, a vast number of small regions of the crystal can be givendifferent conductivities In addition, some dopants establish the electron carrier density(free electron density) while others establish the “hole” carrier density (holes are the dual ofelectrons within semiconductors) In this manner, different types of semiconductor (n-typewith much higher electron carrier density than the hole density and p-type with much higherhole carrier density than the electron carrier density) can be located in small but contactingregions within the crystal.

By applying electric fields appropriately, small regions of the semiconductor can beplaced in a state in which all the carriers (electron and hole) have been expelled by theelectric field, and that electric field sustained by the exposed dopant ions This allows electricswitching between a conducting state (with a settable resistivity) and a non-conducting state(with conductance vanishing as the carriers vanish)

This combination of localized regions with precisely controlled resistivity (dominated byelectron conduction or by hole conduction) combined with the ability to electronically controlthe flow of the carriers (electrons and holes) leads to the semiconductors being the foundationfor contemporary electronics This foundation is particularly strong because a wide variety ofatomic elements (and mixtures of atomic elements) can be used to tailor the semiconductormaterial to specific needs The dominance of silicon semiconductor material in the electronicsarea (e.g., the VLSI digital electronics area) contrasts with the rich variety of semiconductormaterials widely used in optoelectronics In the latter case, the ability to adjust the bandgap

to desired wavelengths of light has stimulated a vast number of optoelectronic components,based on a variety of technologies Electronic components also provide a role for non-siliconsemiconductor technologies, particularly for very high bandwidth circuits which can takeadvantage of the higher speed capabilities of semiconductors using atomic elements similar tothose used in optoelectronics This rich interest in non-silicon technologies will undoubtedlycontinue to grow, due to the rapidly advancing applications of optoelectronics, for the simple

reason that silicon is not suitable for producing an efficient optical source

This chapter provides an overview of many of the semiconductor materials in use Toorganize the information, the topic is developed from the perspective of the factors contribut-ing to the electronic behavior of devices created in the semiconductor materials Section 2discusses the underlying crystalline structure and the semiconductor parameters which resultfrom that structure Section 3 discusses the energy band properties in more detail, extract-ing the basic semiconductor parameters related to those energy bands Section 4 discussescarrier transport Crystalline defects, which can profoundly impact the behavior of devicescreated in the semiconductors, are reviewed in Section 5

2 Crystalline Structures

2.1 Basic Semiconductor Materials Groups

Most semiconductor materials are crystals created by atomic bonds through which the lence band of the atoms are filled with 8 electrons through sharing of an electron from each offour nearest neighbor atoms These materials include semiconductors composed of a singleatomic species, with the basic atom having four electrons in its valence band (supplemented

va-by covalent bonds to four neighboring atoms to complete the valence band) These elementalsemiconductors therefore use atoms from group IV of the atomic chart Other semiconduc-tor materials are composed of two atoms, one from group N (N < 4) and the other from

group M (M > 4) with N + M = 8, filling the valence bands with 8 electrons The major

categories of semiconductor material are summarized below

2.1.1 Elemental (IV-IV) Semiconductors

Elemental semiconductors consist of crystals composed of only a single atomic element from

group IV of the periodic chart, i.e., germanium (Ge), silicon (Si), carbon (C), and tin (Sn).Silicon is the most commonly used electronic semiconductor material, and is also the mostcommon element on earth Table 1 summarizes the naturally occurring abundance of someelements used for semiconductors, including non-elemental (compound) semiconductors

Table 1: Abundance (fraction of elements occurring on earth) of common elements used forsemiconductors

SixGe1−xsemiconductors are under present study to achieve bandgap engineering within thesilicon system In this case, a fraction x (0 < x < 1) of the atoms in an otherwise silicon

crystal are silicon while a fraction 1− x have been replaced by germanium This ability to

replace a single atomic element with a combination of two atomic elements from the samecolumn of the periodic chart appears in the other categories of semiconductor describedbelow (and is particularly important for optoelectronic devices)

Figure 1: Bonding arrangements of atoms in semiconductor crystals (a) Elemental conductor such as silicon (b) Compound III-V semiconductor such as GaAs (c) CompoundII-VI semiconductor such as CdS

The III-V semiconductors are prominent (and will gain in importance) for applications ofoptoelectronics In addition, III-V semiconductors have a potential for higher speed opera-tion than silicon semiconductors in electronics applications, with particular importance for

areas such as wireless communications The compound semiconductors have a crystal

lat-tice constructed from atomic elements in different groups of the periodic chart The III-Vsemiconductors are based on an atomic element A from Group III and an atomic element

B from Group V Each Group III atom is bound to four Group V atoms, and each Group

V atom is bound to four Group III atoms, giving the general arrangement shown in Figure1b The bonds are produced by sharing of electrons such that each atom has a filled (8

electron) valence band The bonding is largely covalent, though the shift of valence chargefrom the Group V atoms to the Group III atoms induces a component of ionic bonding tothe crystal (in contrast to the elemental semiconductors which have purely covalent bonds).Representative III-V compound semiconductors are GaP, GaAs, GaSb, InP, InAs, and InSb.GaAs is probably the most familiar example of III-V compound semiconductors, usedfor both high speed electronics and for optoelectronic devices Optoelectronics has takenadvantage of ternary and quaternary III-V semiconductors to establish optical wavelengths

and to achieve a variety of novel device structures The ternary semiconductors have the

general form (Ax, A 01−x)B (with two group III atoms used to fill the group III atom positions

in the lattice) or A(Bx, B10 −x) (using two group V atoms in the Group V atomic positions

in the lattice) The quaternary semiconductors use two Group III atomic elements and

two Group V atomic elements, yielding the general form (Ax, A 01−x)(By, B10 −y) In suchconstructions, 0 ≤ x ≤ 1 Such ternary and quaternary versions are important since the

mixing factors (x and y) allow the bandgap to be adjusted to lie between the bandgaps

of the simple compound crystals with only one type of Group III and one type of Group

V atomic element The adjustment of wavelength allows the material to be tailored forparticular optical wavelengths, since the wavelength λ of light is related to energy (in this

case the gap energy Eg) by λ = hc/Eg, where h is Plank’s constant and c is the speed

of light Table 2 provides examples of semiconductor laser materials and a representativeoptical wavelength for each, providing a hint of the vast range of combinations which areavailable for optoelectronic applications Table 3, on the other hand, illustrates the change

in wavelength (here corresponding to color in the visible spectrum) by adjusting the mixture

of a ternary semiconductor

Table 2: Semiconductor optical sources and representative wavelengths

Material layers used wavelength

Table 3: Variation of x to adjust wavelength in GaAsxP1−x semiconductors

Ternary Compound ColorGaAs0.14P0.86 YellowGaAs0.35P0.65 Orange

FCC Lattice A

FCC Lattice B

These semiconductors are based on one atomic element from Group II and one atomic ement from Group VI, each type being bonded to four nearest neighbors of the other type

el-as shown in Figure 1c The increel-ased amount of charge from Group VI to Group II atomstends to cause the bonding to be more ionic than in the case of III-V semiconductors II-VIsemiconductors can be created in ternary and quaternary forms, much like the III-V semi-conductors Although less common than the III-V semiconductors, the II-VI semiconductorshave served the needs of several important applications Representative II-VI semiconductorsare ZnS, ZnSe,and ZnTe (which form in the zinc blende lattice structure discussed below);CdS and CdSe, (which can form in either the zinc blende or the wurtzite lattice structure)and CdTe which forms in the wurtzite lattice structure

2.2 Three-Dimensional Crystal Lattice

The two-dimensional views illustrated in the previous section provide a simple view of thesharing of valence band electrons and the bonds between atoms However, the full 3-D latticestructure is considerably more complex than this simple 2-D illustration Fortunately, mostsemiconductor crystals share a common basic structure, developed below

Figure 2: Three-dimensional crystal lattice structure (a) Basic cubic lattice (c) centered cubic (fcc) lattice (c) Two interpenetrating fcc lattices In this figure, the dashedlines between atoms are not atomic bonds but instead are used merely to show the basicoutline of the cube

Face-The crystal structure begins with a cubic arrangement of 8 atoms as shown in Figure

2a This cubic lattice is extended to a face-centered cubic (fcc) lattice, shown in 2b, by

adding an atom to the center of each face of the cube (leading to a lattice with 14 atoms)

The lattice constant is the side dimension of this cube.

The full lattice structure combines two of these fcc lattices, one lattice interpenetratingthe other (i.e., the corner of one cube is positioned within the interior of the other cube,with the faces remaining parallel), as illustrated in Figure 2c For the III-V and II-VIsemiconductors with this fcc lattice foundation, one fcc lattice is constructed from one type

of element (e.g., type III) and the second fcc lattice is constructed from the other type ofelement (e.g., group V) In the case of ternary and quaternary semiconductors, elementsfrom the same atomic group are placed on the same fcc lattice All bonds between atomsoccur between atoms in different fcc lattices For example, all Ga atoms in the GaAs crystalare located on one of the fcc lattices and are bonded to As atoms, all of which appear onthe second fcc lattice The interatomic distances between neighboring atoms is therefore lessthan the lattice constant For example, the interatomic spacing of Si atoms is 2.35 ˚ A but

the lattice constant of Si is 5.43 ˚ A.

If the two fcc lattices contain elements from different groups of the periodic chart,

the overall crystal structure is called the zinc blende lattice In the case of an elemental

semiconductor such as silicon, silicon atoms appear in both fcc lattices and the overall

crystal structure is called the diamond lattice (carbon crystallizes into a diamond lattice

creating true diamonds, and carbon is a group IV element) As in the discussion regardingIII-V semiconductors above, the bonds between silicon atoms in the silicon crystal extendbetween fcc sublattices

Although the common semiconductor materials share this basic diamond/zinc blendelattice structure, some semiconductor crystals are based on a hexagonal close-packed (hcp)lattice Examples are CdS and CdSe In this example, all the Cd atoms are located onone hcp lattice while the other atom (S or Se) is located on a second hcp lattice In thespirit of the diamond and zinc blende lattices above, the complete lattice is constructed by

interpenetrating these two hcp lattices The overall crystal structure is called a wurtzite

lattice Type IV-VI semiconductors (PbS, PbSe, PbTe, and SnTe) exhibit a narrow bandgap and have been used for infrared detectors The lattice structure of these example IV-VI

semiconductors is the simple cubic lattice (also called an NaCl lattice).

2.3 Crystal Directions and Planes

Crystallographic directions and planes are important in both the characteristics and the plications of semiconductor materials since different crystallographic planes can exhibit sig-nificantly different physical properties For example, the surface density of atoms (atoms/cm2)can differ substantially on different crystal planes A standardized notation (the so-called

ap-Miller indices) is used to define the crystallographic planes and directions normal to those

planes

The general crystal lattice defines a set of unit vectors (a,b, and c) such that an entire

crystal can be developed by copying the unit cell of the crystal and duplicating it at integeroffsets along the unit vectors, i.e., replicating the basis cell at positions naa +nbb +ncc,

where na, nb, and nc are integers The unit vectors need not be orthogonal in general For

Su-plane’s intercept with the x-axis, k corresponds to the plane’s intercept with the y-axis and

l corresponds to the plane’s intercept with the z-axis Since parallel planes are equivalent

planes, the intercept integers are reduced to the set of the three smallest integers having thesame ratios as the above intercepts The (100), (010) and (001) planes correspond to thefaces of the cube The (111) plane is tilted with respect to the cube faces, intercepting the

x, y, and z axes at 1, 1, and 1, respectively In the case of a negative axis intercept, the

corresponding Miller index is given as an integer and a bar over the integer,e.g., (¯100), i.e.,similar to (100) plane but intersecting x-axis at -1

Figure 3: Examples of crystallographic planes within a cubic lattice organized semiconductorcrystal (a) (010) plane (b) (110) plane (c) (111) plane

Additional notation is used to represent sets of planes with equivalent symmetry and torepresent directions For example,{100} represents the set of equivalent planes (100), (¯100).

(010), (0¯10), (001), and (00¯1) The direction normal to the (hkl) plane is designated [hkl].

The different planes exhibit different behavior during device fabrication and impact electricaldevice performance differently One difference is due to the different reconstructions of thecrystal lattice near a surface to minimize energy Another is the different surface density ofatoms on different crystallographic planes For example, in Si the (100), (110), and (111)planes have surface atom densities (atoms per cm2) of 6.78 ×1014, 9.59 ×1014, and 7.83 ×1014,respectively

A semiconductor crystal establishes a periodic arrangement of atoms, leading to a periodicspatial variation of the potential energy throughout the crystal Since that potential energy

varies significantly over interatomic distances, quantum mechanics must be used as the basisfor allowed energy levels and other properties related to the semiconductor Different semi-conductor crystals (with their different atomic elements and different inter-atomic spacings)lead to different characteristics However, the periodicity of the potential variations leads

to several powerful general results applicable to all semiconductor crystals Given thesegeneral characteristics, the different semiconductor materials exhibit properties related tothe variables associated with these general results A coherent discussion of these quantummechanical results is beyond the scope of this chapter and we therefore take those generalresults as given

In the case of materials which are semiconductors, a central result is the momentum functions defining the state of the electronic charge carriers In addition to

energy-the familiar electrons, semiconductors also provide holes (i.e positively charged particles)

which behave similarly to the electrons Two energy levels are important: one is the energy

level (conduction band) corresponding to electrons which are not bound to crystal atoms and which can move through the crystal and the other energy level (valence band) corresponds to

holes which can move through the crystal Between these two energy levels, there is a region

of “forbidden” energies (i.e., energies for which a free carrier can not exist) The separation

between the conduction and valence band minima is called the energy gap or band gap The

energy bands and the energy gap are fundamentally important features of the semiconductormaterial and are reviewed below

In quantum mechanics, a “particle” is represented by a collection of plane waves (e j(ωt −~k·~x))where the frequency ω is related to the energy E according to E = ¯ hω and the momentum

p is related to the wave vector by ~ p = ¯ h~k In the case of a classical particle with mass m

moving in free space, the energy and momentum are related by E = p2/(2m) which, using

the relationship between momentum and wave vector, can be expressed asE = (¯ hk)2/(2m).

In the case of the semiconductor, we are interested in the energy/momentum relationship for

a free electron (or hole) moving in the semiconductor, rather than moving in free space Ingeneral, this E-k relationship will be quite complex and there will be a multiplicity of E-k

“states” resulting from the quantum mechanical effects One consequence of the periodicity

of the crystal’s atom sites is a periodicity in the wave vector k, requiring that we consider

only values of k over a limited range (with the E-k relationship periodic in k).

Figure 4 illustrates a simple example (not a real case) of a conduction band and a valenceband in the energy-momentum plane (i.e., the E vs k plane) The E vs k relationship of theconduction band will exhibit a minimum energy value and, under equilibrium conditions,the electrons will favor being in that minimum energy state Electron energy levels abovethis minimum (Ec) exist, with a corresponding value of momentum The E vs k relationshipfor the valence band corresponds to the energy-momentum relationship for holes In thiscase, the energy values increase in the direction toward the bottom of the page and the

“minimum” valence band energy level Ev is the maximum value in Figure 4 When anelectron bound to an atom is provided with sufficient energy to become a free electron, a

Γ Κ

kk

Conduction band minimum (free electrons)

Valence band minimum (free holes)

Energy gap Eg

Figure 4: General structure of conduction and valence bands

hole is left behind Therefore, the energy gap Eg =Ec − Ev represents the minimum energynecessary to generate an electron-hole pair (higher energies will initially produce electronswith energy greater than Ec, but such electrons will generally lose energy and fall into thepotential minimum)

The details of the energy bands and the bandgap depend on the detailed quantummechanical solutions for the semiconductor crystal structure Changes in that structure(even for a given semiconductor crystal such as Si) can therefore lead to changes in theenergy band results Since the thermal coefficient of expansion of semiconductors is non-zero, the band gap depends on temperature due to changes in atomic spacing with changingtemperature Changes in pressure also lead to changes in atomic spacing Though thesechanges are small, the are observable in the value of the energy gap Table 4 gives the roomtemperature value of the energy gap Eg for several common semiconductors, along with therate of change of Eg with temperature (T ) and pressure (P ) at room temperature.

The temperature dependence, though small, can have a significant impact on carrierdensities A heuristic model of the temperature dependence of Eg is Eg(T ) = Eg(0o K) −

αT2/(T + β) Values for the parameters in this equation are provided in Table 5 Between

0K and 1000K, the values predicted by this equation for the energy gap of GaAs are accurate

to about 2× 10 −3 eV (electron volts)

Energy gap

Conduction

band minimum

Valence band minimum

3.2 Direct Gap and Indirect Gap Semiconductors

Figure 5 illustrates the energy bands for Ge, Si and GaAs crystals In Figure 5b, for silicon,

the valence band has a minimum at a value of ~ k different than that for the conduction band

minimum This is an indirect gap, with generation of an electron-hole pair requiring an

energy Eg and a change in momentum (i.e., k) For direct recombination of an electron-hole

pair, a change in momentum is also required This requirement for a momentum change (incombination with energy and momentum conservation laws) leads to a requirement that aphonon participate with the carrier pair during a direct recombination process generating

a photon This is a highly unlikely event, rendering silicon ineffective as an optoelectronicsource of light The direct generation process is more readily allowed (with the simultaneousgeneration of an electron, a hole, and a phonon), allowing silicon and other direct gapsemiconductors to serve as optical detectors

Figure 5: Conduction and valence bands for (a) germanium, (b) silicon, and (c) GaAs.Adapted from [4]

In Figure 5c, for GaAs, the conduction band minimum and the valence band minimum

occur at the same value of momentum, corresponding to a direct gap Since no momentum

change is necessary during direct recombination, such recombination proceeds readily, ducing a photon with the energy of the initial electron and hole (i.e., a photon energy equal

pro-to the bandgap energy) For this reason, direct gap semiconducpro-tors are efficient sources oflight (and use of different direct gap semiconductors with different Eg provides a means oftailoring the wavelength of the source) The wavelength λ corresponding to the gap energy

is λ = hc/Eg

Figure 5c also illustrates a second conduction band minimum with an indirect gap, but

at a higher energy than the minimum associated with the direct gap The higher conductionband minimum can be populated by electrons (which are in an equilibrium state of higherenergy) but the population will decrease as the electrons gain energy sufficient to overcomethat upper barrier

3.3 Effective Masses of Carriers

For an electron with energy close to the minimum of the conduction band, the energy vsmomentum relationship is approximately given byE(k) = E0+a2(k −k ∗)2+a4(k −k ∗)4+ .

Here, E0 =Ec is the “ground state energy” corresponding to a free electron at rest andk ∗ isthe wave vector at which the conduction band minimum occurs Only even powers of k − k ∗

appear in the expansion of E(k) around k ∗ due to the symmetry of the E-k relationshiparound k = k ∗ The above approximation holds for sufficiently small increases in E above

Ec For sufficiently small movements away from the minimum (i.e., sufficiently small k −k ∗),the terms ink −k ∗ higher than quadratic can be ignored andE(k) ≈ E0+a2k2, where we havetakenk ∗ = 0 If, instead of a free electron moving in the semiconductor crystal, we had a freeelectron moving in free space with potential energy E0, the energy-momentum relationshipwould be E(k) = E0+ (¯hk)2/(2m0), where m0 is the mass of an electron By comparison ofthese results, it is clear that we can relate the curvature coefficient a2 associated with the

parabolic minimum of the conduction band to an effective mass m ∗ e, i.e.,a2 = (¯h2)/(2m ∗ e) or

Since the energy bands depend on temperature and pressure, the effective masses can also

be expected to have such dependencies, though the room temperature and normal pressurevalue is normally used in device calculations

The above discussion assumes the simplified case of a scalar variablek In fact, the wave

vector ~ k has three components (k1, k2, k3), with directions defined by the unit vectors of theunderlying crystal Therefore, there are separate masses for each of these vector components

of ~ k, i.e., masses m1, m2, m3 A scalar massm ∗ can be defined using these directional masses,the relationship depending on the details of the directional masses For cubic crystals (as inthe diamond and zinc blende structures), the directions are the usual orthonormal directionsandm ∗ = (m1·m2·m3)1/3 The three directional masses effectively reduce to two components

if two values are equal (e.g.,m1 =m2), as in the case of longitudinal and transverse effective

masses (ml and mt, respectively) seen in silicon and several other semiconductors In thiscase, m ∗ = [(mt)2· ml]1/3 If all three values of m1, m2, m3 are equal, then a single value m ∗

can be used

An additional complication is seen in the valence band structures in Figure 5 Here,two different E-k valence bands have the same minima Since their curvatures are different,

the two bands correspond to different masses, one corresponding to heavy holes with mass

mh and the other to light holes with mass ml The effective scalar mass in this case is

m ∗ = (m3h /2+m3l /2)2/3 Such light and heavy holes occur in several semiconductors, including

Si

Values of effective mass are given in Tables 8 and 13

3.4 Intrinsic Carrier Densities

The density of free electrons in the conduction band depends on two functions One is thedensity of states D(E) in which electrons can exist and the other is the energy distribution

function F (E, T ) of free electrons.

The energy distribution function (under thermal equilibrium conditions) is given by theFermi-Dirac distribution function

F (E) =

·

1 + exp

µE − Ef kBT

¶¸−1

which, in most practical cases, can be approximated by the classical Maxwell-Boltzmanndistribution These distribution functions are general functions, not depending on the specificsemiconductor material

The density of statesD(E), on the other hand, depends on the semiconductor material.

for free holes due to the forbidden region between Ec and Ev

The density n of electrons in the conduction band can be calculated as

n =

Z ∞

E=E c F (E, T )D(E)dE.

For Fermi levels significantly (more than a fewkBT ) below Ec and aboveEv, this integrationleads to the results

n = Nce −(Ec−Ef)/k b T and

p = Nve −(Ef −Ev)/k b T

where n and p are the densities of free electrons in the conduction band and of holes in the

valence band, respectively NcandNv are effective densities of states which vary with

temper-ature (slower than the exponential in the equations above), effective mass, and other tions Table 6 gives values ofNc andNv for several semiconductors Approximate expressionsfor these densities of state are Nc = 2(2πm ∗ e kBT /¯ h2)3/2 Mc and Nv = 2(2πm ∗ e kBT /¯ h2)3/2 Mv.These effective densities of states are fundamental parameters used in evaluating the elec-trical characteristics of semiconductors The equations above for n and p apply both to intrinsic semiconductors (i.e., semiconductors with no impurity dopants) as well as to semi-

condi-conductors which have been doped with donor and/or acceptor impurities Changes in theinterrelated values of n and p through introduction of dopant impurities can be represented

by changes in a single variable, the Fermi level Ef

The product of n and p is independent of Fermi level and is given by

n · p = Nc · Nve −Eg /k B T

where the energy gap Eg = Ec − Ev Again, this holds for both intrinsic semiconductorsand for doped semiconductors In the case of an intrinsic semiconductor, charge neutralityrequires that n = p ≡ ni, where ni is the intrinsic carrier concentration and

n2i =Nc · Nve −Eg/k B T

Since, under thermal equislibrium conditions np ≡ n2

i (even under impurity doping tions), knowledge of the density of one of the carrier types (e.g., of p) allows direct deter-

condi-mination of the density of the other (e.g., n = n2

i /p) Values of ni vary considerably amongsemiconductor materials: 2× 10 −3 /cm3 for CdS, 3.3 × 106/cm3 for GaAs, 0.9 × 1010/cm3 for

Si, 1.9 × 1013/cm3 for Ge, and 9.1 × 1014 for PbS

Since there is appreciable temperature dependence in the effective density of states,the equations above do not accurately represent the temperature variations in ni over widetemperature ranges Using the approximate expressions for Nc and Nv above,

a variety of dopants and their energy levels for Si and GaAs.

Table 7: Acceptor and donor impurities used in Si and GaAs Adapted from [1]

Donor Ec − Ed (eV) Acceptor Ea − Ev (eV)