The Physics of Semiconductor Materials

The Physics of Semiconductor Materials ELIAS BURSTEIN

AND

PAUL H. EGLI

Crystal Branch United States Naval Research Laboratory Washington, D.C.

Page 1 I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ 3 11. Nature of Semiconductors. . . . . . . . . . 1. Chemical Binding.. . . . . . . . . . . . . . ................ 3 8 2. Energy Band Structures.. 18 111. Trends in Fundamental Prop 20 1. Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2. Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 IV. Impurity and Lattice Defect Centers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1. Character of Levels in the Forbidden Band. 2. Homopolar Semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3. Polar Semiconductors ..... . . . . . . . . . . . . . . . . 45 V. Lifetime of Electron-Hol ...................................... 52 VI. Procedures for Determining Characteristic Properties of Semiconductors. . . 59 1. Carrier Concentration and Mobility.. ....................... 2. Forbidden Energy Gap.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Energy Levels within the Forbidden Band ..................... 65 4. Effective Mass.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 VII. Current Information on Familiar Semiconductor Materials. . . . . . . . . . . . . . 69 1. Elemental Semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ................................. 73 2. Compound Semiconductors References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

I. INTRODUCTION Semiconductors have been the focal point for a vast amount of progress in many areas of science and technology. The development of new devices requiring a variety of properties from crystals has stimulated both the search for new materials and a more intensive study of the properties of long familiar semiconductors. The definition of a semiconductor, and hence the materials t o be included in this discussion, must be somewhat arbitrary. The distinction between semiconductors and metals is clear cut. The distinction, however, between semiconductors and insulators (or " dielectrics ") is largely a matter of degree. The materials of interest for a thermoelectric device, for example, are quite different from those required for rectifiers, although 1

2

ELIAS BURSTEIN AND PAUL H . EGLI

both are properly regarded as semiconductors. Most of the current interest is in some form of rectifier, amplifier, or photoconductor; and all of these share certain common property requirements. Each of these applications requires an energy gap within a limited range, and charge carrier mobilities and lifetimes beyond certain minimum values. These basic properties, which characterize a semiconductor, show trends which are related to the structure and type of chemical binding of the solid. By consideration of these trends in various families of materials, it is possible to establish some of the boundaries concerning crystals which will be of interest and what properties may be expected. What appeared a t first to be a limitless sea of semiconductor materials can thus be reduced to a very small number as soon as the property requirements are established. Considerable progress has been made in the preparation of materials, particularly with respect to purification by the zone methods and with respect to stoichiometry control; but inherent preparation difficulties still add a further restriction t o the number of practical materials. There has been, in the field of semiconductors, an unusually effective interchange between research and development. The potential impact of transistors in electronics has stimulated an enormous research effort ;and the information thus acquired has been rapidly applied. Development of new tools such as cyclotron resonance, and the full exploitation of optical phenomena, to name only two of the many approaches, has produced a mass of new information. From such works have come significant contributions covering a broad spectrum of the physics of solids. Notable progress has been made, for example, in clarifying the structure of the valence and conduction bands. Details concerning the energy gap, e.g. the role of nonvertical transitions, and the distinctions between optical and thermal gaps is another area where progress has been rapid. Valuable contributions have been made toward clarifying the character of the energy levels in the forbidden zone, the role of impurity and defect centers. Many other contributions deserve attention, for example the discovery and explanation of the high mobilities in compound semiconductors. Progress has been notable in clarifying the mechanism of the several recombination processes, and the distinctions between diffusion and conductivity lifetimes. All these, and other contributions to the basic physics of solids are an integral part of the progress derived from the work on semiconductor materials. Both the data and the concepts here presented willundoubtedly be improved, but progress has already been sufficient to make a review of the status of the physics of semiconductor materials seem appropriate (I).

PHYSICS OF SEMICONDUCTOR MATERIALS

3

11. NATURE OF SEMICONDUCTORS

I. Chemical Binding The character of the energy bands, and therefore the fundamental properties of a semiconductor, reflect to a large extent the nature of the chemical binding between the atoms. It may therefore be of value t o review the major types of chemical binding in nonmetallic crystals and to discuss qualitatively the characteristics of each type. a. Character of Binding Types (2). The character of solids with van der Waals binding is illustrated by crystals of the rare gases. I n the solid rare gases, the outer electrons of the atoms are strongly localized, i.e., the wave functions of the valence electrons do not overlap appreciably with those of neighboring atoms. The interatomic distance between the atoms is determined by a balance between the van der Waals attractive forces and the repulsive forces arising from Pauli exclusion effects between atoms with completed outer shells. The filled bands which arise from the ground levels of isolated atoms are accordingly quite narrow. The wave functions associated with the excited states of the atoms exhibit somewhat more overlapping between neighboring atoms. The exciton bands and the conduction band will accordingly be appreciably broader than the filled band. The solid rare gases crystallize in the face centered cubic structure in which the atoms have a coordination number of 6. They are nonpolar and are characterized by low binding energies, low melting points, and large energy gaps. The alkali halides typify crystals in which the binding is predominantly ionic. They are made up of positive and negative ions which have completed outer electronic shells formed by the transfer of the valence electrons of the strongly electropositive alkali metal atoms t o the strongly electronegative halogen atoms. The crystals are held together primarily by the strong electrostatic attraction between the ions. The interatomic distance is essentially determined by a balance between this coulombic attractive force and repulsive forces of the type exhibited by atoms with completed outer shells. Since the free ions have rare gas configurations, the outer electrons are strongly localized. However, as a result of the strong coulombic attraction the wave function of the valence electrons on neighboring ions overlap appreciably more than they do in the case of the solid rare gases. The filled bands are consequently appreciably broader than those of the solid rare gases, but there is still very little mixing of the halogen and the alkali metal ion energy levels (Fig. 1). The ionic crystals exhibit a strong infrared lattice vibration absorption

4

ELIAS BURSTEIN AND PAUL H. EGLI

and an appreciable difference between the optical and static dielectric constants. They are also characterized by high binding energies, high melting points, and relatively large energy gaps. The simple ionic crystals generally crystallize in NaCl and CsCl structures in which the ions have coordination numbers of 6 and 8, respectively; or in the CaFz structure in which the coordination numbers of the cations and anions are 8 and 4, respectively.

Interatomic distance

FIG.1. The dependence of the energy bands of NaCl on interatomic distance. The equilibrium interatomic distance is shown at d [J. C . Slater and W. Shockley, Phys. Rev. 60, 705 (1936)l.

The transition metal oxides such as MnO, COO, and NiO constitute another group of crystals which exhibit predominantly ionic binding. In these substances the cations have incompletely filled 3d shells (Mn+2 = 3d6, C O + ~= 3d7, Ni+2 = 3da). The bands formed from the degenerate 3d levels are, however, quite narrow since the wave function of the 3d electrons are highly localized, and are believed to lie above the bands formed from the oxygen levels. These compounds crystallize in the NaCl structure. They, like the alkali halides, are characterized by high binding energies and melting points and by relatively large energy gaps. The group IV-B elements-diamond, silicon, germanium, and gray tin,-and the MII1-NV*compounds are examples of solids having predominantly covalent binding. The atoms in these crystals are each bound to four nearest neighbors by tetrahedral covalent bonds, in which the

* M is used to denote the more metallic, and N the less metallic component of the compound.

PHYSICS OF SEMICONDUCTOR MATERIALS

5

valence electrons in pairs are shared by neighboring atoms. The allowed energy bands arise from combinations of the s and p valence electron levels of the isolated atoms and are generally broader than those of the van der Waals and ionic crystals (Fig. 2). The highly covalent crystals are nonpolar, or only weakly polar, and exhibit little or no differencebetween the optical and static dielectric constants. They are characterized by high binding energies and high melting points. Their energy gaps range from values on the order of 5 ev t o values below 0.1 ev. The “simple”

Conduction band

-0.5.

+ .-

2P

v)

5

-1.0.

:

2s

n

-

R f x -1.5. w

-2.0-2.5. 1

2

3

4

5

6

R (Atomic units)

FIQ.2. The dependence of the energy bands of diamond on interatomic distance [G. F. Kinball, J. Chem. Phys. 3, 560 (193511.

covalent crystals with cubic structure generally crystallize in the zinc blende or diamond structure in which the atoms have a coordination number of 4. b. Periodic Trends in Binding. It would be highly desirable to be able to predict properties of new semiconductor materials on the basis of the type of binding to be expected; and to a limited extent this is possible. The type of binding and chemical properties exhibited by simple binary ( M - N ) compounds on the basis of the periodic relation of atoms has been the subject of many treatises (3). Beyond the general rules for compound formation based on differences in electronegativity, atomic number, position on a periodic chart, atomic radii, etc., there are certain other clues as to type of binding based on additional information which is usually available. For example, a phase equilibrium study of the system in question frequently provides valuable information. Additional evidence of the type of binding can be deduced from a knowledge of the structure of the compound. A smooth continuous gradation from ionic to homopolar binding is disrupted by the fact that the atoms in a solid must be packed in one of a limited number of symmetrical configurations. This packing arrange-

6

ELIAS BURSTEIN AND PAUL H. EGLI

ment provides a clue as t o the amount of ionic and homopolar character which can be expected. Thus before attempting t o relate the type of binding to the particular properties of interest in semiconductors, it is desirable to review the trends in binding and structure found for the nonmetallic elements and the more prominent families of compounds with simple structures. Trends in Bindine of Elements. The elements which form nonmetallic crystals include the lower atomic number elements of groups VII-B,

Pl

n IA

,? ip M A IVA V & K C20a S 21c z T ; I P

19

Rb i r Y

Zr b!

C s 56Ba 57La 72Hf 73l a

35

FIG.3. Abbreviated form of the periodic table of the elements.

VI-B, V-B, and IV-B of the periodic table (Fig. 3). These elements lack 1, 2, 3, and 4 electrons, respectively, in their outer shells and tend t o form covalent bonds with (8 - N ) nearest neighbors. Thus the group VII-B elements form crystals made up of diatomic molecules. The group VI-B elements tend to form chains in which each atom has two neighbors, the group V-B elements tend to form structures in which each atom is bound t o three neighbors, and the group IV-B elements tend t o form the diamond structure in which each atom is bound t o four nearest neighbors. With increasing atomic number there is an increasing tendency for the elements of these four groups t o form metals. This tendency is associated with the decreasing ionization energies of the valence electrons and the consequent decrease in binding energy of the covalent bonds. The metallic forms of these elements generally exhibit covalent as well as metallic binding. Thus in Sb and Bi and the metallic form of As, the structure consists of layers of atoms in which the atoms are bound by covalent bonds t o three nearest neighbors. Trends in Binding of Compounds. I n forming compounds, the periodicity of the elements also leads t o well-defined trends. I n going from compounds with classical ionic binding to compounds with classical co-

PHYSICS OF SEMICONDUCTOR MATERIALS

7

valent binding the structures shift from sodium chloride and cesium chloride t o wurtzite* t o zinc blende t o diamond. The elements from periods I-A, 11-A, and 111-A tend t o form ionic compounds when combining with the electronegative elements. Thus the strongly electropositive group I-A alkali metal elements and the strongly electronegative group VII-B halogen elements combine t o form highly having NaCl and CsCl structures. ionic compounds (M1-A-NV1l-B) TABLEI* Structure of

cu Ag

compounds

F

c1

Br

I

z

Z R

Z R

Z W,Z

R Structure of

0

Structure of

A1 Ga In

MI-B-NV11-B

MII-B-NVI-B

S

M"1-B-NV-B

compounds Se

Te

compounds

N

P

As

Sb

W W W

Z Z Z

Z Z

Z Z

Z

Z

* R indicates the rocksalt, Z the zinc blende and W the wurtzite structure. The existence of other structures is indicated by 0. The IIP1-A-NVx-B compounds also form sodium chloride structures but, since they are less strongly ionic, they also form wurtzite and zinc blende structures. The group I-B, 11-B, and 111-B metallic elements combine with the electronegative group V-B, VI-B, and VII-B elements t o form more highly covalent compounds with wurtzite or zinc blende structures (Table I). Of

* The wurtzite and zinc blende structures differ only in the position of an atom relative to its third nearest neighbors, which are of opposite charge. This distance is smaller in the wurtzite structure than in the zinc blende structure. The wurtzite structure is therefore favored by ionic binding.

8

ELIAS BURSTEIN AND PAUL H . EGLI

these families of compounds, the M1-B-NV1r-B group are appreciably ionic and some members occur in sodium chloride structure. The M1l-B-NV1-B group are less ionic and predominantly occur in wurtzite structures. The M1rl-B-NV-B group have the least difference in electronegativity and are the most covalent; and with increasing atomic number shift from wurtzite structure to zinc blende. Still more highly covalent compounds are formed when the group Sic still IV-B elements combine with one another t o form MIV-B-NIV-B. has an appreciable ionic character, and crystallizes in either the wurtzite or the zinc blende structure. Other members of the group, however, C-C, Si-Si, Ge-Gel and Sn-Sn, which may also be regarded as being compounds in which M = N , are the most perfectly covalent compounds. The group IV-B elements, Sn and Pb, and the group 111-B element T1 exhibit more than one valence. Accordingly, we find that they behave like the group A elements in compounds where they exhibit the smaller of their valences. Thus TI combines with the halogens to form ionic compounds with CsCl structure, and P b combines with s, Se, and Te t o form relatively ionic compounds with NaCl structures.

2. Energy Band Structures Two types of approximations have been employed in calculating the wave function of outer electrons of atoms in crystals: the Heitler-London atomic or “close coupling” approximation, and the Bloch band or “collective electron” approximation (4). In the atomic approximation it is assumed that the valence electrons are tightly bound to the atoms and that their wave functions are localized. I n the band approximation, on the other hand, it is assumed that the valence electrons are associated with the entire lattice and that their wave functions extend over the entire lattice. The atomic approximation can be used to calculate the allowed energy states in the valence bands of crystals such as the solid rare gases and alkali halides which are made up of atoms or ions with rare gas configurations. It cannot, however, be readily used t o calculate the allowed energy states of electrons in the conduction bands of crystals, nor can it be readily used for calculating the allowed energy states in the valence bands of crystals having predominantly covalent binding. The band approximation, on the other hand, can be used for calculating the allowed energy states in the conduction band as well as in the valence band of ionic and covalent crystals and is generally the more useful one for interpreting the properties of semiconductors. According to the band approximation, the allowed energy states of the electrons are multivalued functions of their momenta and are conveniently described in terms of energy-momentum curves. These energy-

PHYSICS OF SEMICONDUCTOR MATERIALS

9

momentum curves which correspond to the allowed energy bands of the crystal have discontinuities at the Brillouin zone boundaries which are characteristic of the crystal structure. The number of states in each allowed energy band is equal to the number of primitive unit cells in the crystal multiplied by two for the two spins of the electron. The ranges of energy for which there are no allowed energy bands constitute the forbidden energy bands of the crystal. I n terms of the band theory, semiconductors and insulators represent substances in which the allowed energy bands are either completely filled or completely empty a t absolute zero, and the highest filled band and the lowest empty band are separated by a finite energy gap. Metals, on the other hand, have incompletely filled allowed energy bands a t absolute zero temperature due either t o a n insufficient number of electrons per unit cell or to a n overlapping of otherwise filled and empty bands. Metals can accordingly conduct current even at absolute zero, since the electrons within the partially filled bands can make transitions from occupied to unoccupied states under the influence of an applied field; whereas semiconductors and insulators are perfect “insulators ” a t absolute zero temperature. The band theory does not, however, adequately describe the properties of the transition metal oxides (e.g., NiO, COO, and MnO) which are nonmetallic and yet have incompletely filled 3d bands. It has therefore been suggested that it is necessary to use the “atomic” rather than the “collective electron” approximation t o describe the electrical properties of these materials (5).According to the Heitler-London point of view, the 3d electrons are highly localized a t the transition metal ions. I n order for conductivity to take place in NiO, for example, an electron must be transferred from an occupied level on one Nif2 ion to an unoccupied level on a neighboring Nif2 ion t o form a Nif’ ion and a Ni+3 ion. The electron a t the Ni+‘ ion and the positive hole at the Ni+3ion can move through the lattice by (‘electronic exchange” with neighboring Ni+2ions. The formation of the Ni+I and the Nif3 ions which have an effective charge of - l e and l e relative t o the average (zero) charge of the surrounding medium involves a separation of charges and therefore requires an energy of the order of several electron volts. a. The Simple Energy Band Model. I n the absence of information about the structure of the energy bands, the properties of semiconductors have been interpreted theoretically in terms of a simple energy band model (Fig. 4) in which the top of the valence band and the bottom of the conduction band are assumed to occur a t a crystal momentum, k = 0, and the energy E of the free electrons and holes is assumed t o be a quadratic function of their momenta k and independent of direction (i.e., the Burfaces of constant energy are assumed t o be spherical) (6). I n this model

+

10

ELIAS RURSTEIN AND PAUL H. EOLI

the effective masses m* of the free carriers which are defined in terms of the curvature at the band edge can be treated as scalar quantities, 1

-

m*

=

-_.

47r2d2E h2 ah2

For semiconductors which can be described in terms of this simple model, the thermal equilibrium concentration of free electrons and holes in the intrinsic range of temperatures under conditions where Boltzman

Momentum

FIG. 4. Energy-momentum curves for the valence and conduction bands of a “simple energy band ” model semiconductor.

statistics apply is determined by the intrinsic ionization equilibrium constant (7)

Ka

= fnfp =

nnnp= N,N, exp (-Eo/lcT),

(2)

where n, and n, are the concentrations of free electrons and holes, fn and f, are the partition functions per unit volume of free electrons and free holes. N , and N , are the effective density of states in the conduction band and valence band which may be expressed in terms of tfheeffective masses m, and m, of electrons and holes. and

N,

=

2(27rm,kT/h2)$5

N,

=

2(27rm,l~T/h~)~~.

(3)

EGis the thermal energy gap which has the properties of a partial Gibbs free energy (8). It is equal t o the forbidden energy gap only when the

PHYSICS OF SEMICONDUCTOR MATERIALS

11

energy levels are independent of temperature, which is never the case in real crystals. According to the simple model, the intrinsic optical properties of semiconductors are determined by the selection rule that, to a good approximation, optical transition of electrons from the valence band t o

K * (0001

K.' 1 112 112 1/21

K

8

11001

l a ) PURE GERMANIUM I

I

I

I b l PURE S I L I C O N t ( l I l 1AXIS

-+-(loo)

AXIS

-1

FIG.5. Schematic diagram of energy bands in germanium and silicon along [loo] and [lll]directions. States normally occupied by electrons and holes at room temperature are shown darkened. The removal of degeneracy by spin-orbit interaction is not shown [F. Herman, Phys. Rev. 96, 847 (1955)l.

the conduction band takes place vertically in order to conserve momentum (9). The optical energy gap, Eo,which corresponds to the minimum photon energy required t o transfer an electron from the valence band to the conduction band should then in principle be equal to the forbidden energy gap. The optical energy gap does however differ from the thermal energy gap because of Franck-Condon effects (10). In ionic crystals where the static dielectric constant e, is appreciably larger than the optical dielectric constant to, the optical energy gap may be appreciably larger than

12

ELIAS BURSTEIN AND PAUL €I EGLI .

the thermal energygap because of polarization effects which depend on the value of (e, - E , J / E ~ C ~ .For homopolar materials, however, where the static dielectric constant is essentially equal to the optical dielectric constant, the difference between the optical and thermal energy gaps will be small. b. Energy Band Structures of Known Materials. It is now known from a variety of experimental evidence (11) that the energy band structures of silicon and germanium are more complicated than has generally been assumed. The known band structures for silicon and germanium based on the cyclotron resonance experiments of Dresselhaus and Lax and their co-workers (12) and theoretical calculations by Herman ( I S ) are shown in Fig. 5 . The top of the valence band and the bottom of the conduction TABLE 11. Energy Gaps and Related Properties of Group IV Elements

Diamond Silicon 1 . 2 1 -4.2 x 10-4 Germanium 0.785 -4.0 X 10-4 Gray Tin 0.08

1.09 0.65 -

5.6 1.05 0.62 -

5 . 7 180 -4.0 X lov4 11.6 146 -4.4 X lo-' 16 159 _ -

* Calculated from the experimental value for EoQand the density of states masses derived from cyclotron resonance data assuming 8 conduction band ellipsoids for germanium and 6 for silicon. band do not appear at the same momentum. The valence band consists of two bands, V1 and V z , which are degenerate at k = 0, and a third band VB which has a lower energy due t o spin-orbit coupling ( i 4 ) . The structure of the conduction band is more complex. For germanium there are eight equivalent minima which occur at points in k space along (111) directions (or possibly four if they occur a t a Brillouin zone boundary). The energy surfaces in the neighborhood of the minima are cigar-shaped ellipsoids with the long axis lying along (111) directions. For silicon, on the other hand, there are six equivalent cigar-shaped ellipsoids along the (100) directions. Herman (16) has suggested that additional minima also occur a t k = 0 and along (100) directions in germanium and at k = 0 and along (111) directions in silicon and has interpreted the dependence of the forbidden energy gap on composition in silicon-germanium alloys (16) as being due to the changein the relative positions of the (100) and (111) minima. Since the minimum vertical separation between the valence and conduction bands is larger than the forbidden energy gap, one would nor-

PHYSICS O F S E M I C O N D U C T O R M A T E R I A L S

13

mally expect the optical energy gaps of silicon and germanium t o be larger than their thermal energy gaps. It is actually found that the energy gap determined from the long wavelength absorption edge is in good agreement with the thermal energy gap (Table 11) (17). Bardeen and co-workers (18) have accordingly suggested that the absorption at the long wavelength edge may involve nonvertical transitions (Fig. 6) which are accompanied by the absorption or emission of phonons in order t o satisfy the requirement of momentum conservation. Such nonvertical transitions which are second order compared t o the vertical transitions

k,

k(ll1)

FIQ.6. Schematic diagram showing vertical (direct) and nonvertical interband transitions in germanium [L. H. Hall, J. Bardccn, and F. J. Blatt, Phys. Rev. 96, 559 (1954)l. would be expected to appear as a tail t o the absorption involving vertical transition. The plausibility of this interpretation is borne out by the optical constant data for silicon and germanium (17,29) (Fig. 7). I n germanium, in going from the absorption edge to shorter wavelengths, there is a n additional small rise in absorption starting at about 0.8 ev and then a more rapid rise at about 1.5 ev which may be attributed t o vertical transitions. I n silicon, there is an additional rise a t about 2.5 ev which may be attributed to direct transitions. Macfarlane and Roberts (17) have recently analyzed the absorption of germanium close to the absorption edge in terms of a nonvertical transition mechanism. They obtain a value for the energy of the phonons involved in the nonvertical transitions between the band edge and k = 0, and on the basis of known values of the elastic constants and the theory of lattice vibrations of the diamond lattice they estimate a momentum value for the position of the conduction band minima which is about N of the momentum a t the zone edge.

14

ELIAS BURSTEIN AND PAUL H. EGLI

Photon energy, electron volts

FIQ.7. Intrinsic absorption spectra of silicon and germanium. The transmission curves are due to W. L. Dash, E. A. Taft, and R. Newman, Bull. Am. Phys. Soe. 30, (V7), 53 (1955); the reflection curve for germanium is based on the data of D. G. Avery and P. L. Clegg, Proc. Phys. SOC.B66, 512 (1953); and the reflection curve for silicon is based on the data of G. Pfestorf, Ann. Phvsik 61, 906 (1926).

-31

I

I

(0001 $(1111 Reduced wave vectorK (Slope of energy contour IS zero at polnts marked “0”)

~.clOol

10001

FIQ.8. Schematic diagram of energy band in diamond along tions [F. Herman, Phys. Rev. 88, 1210 (1952)l.

[loo] and [111] direc-

PHYSICS OF GEMICONDUCTOR MATERIALS

15

Herman has also carried out theoretical calculations of the electronic energy band structure of diamond (80)which indicate that the bottom of the conduction band occurs away from the origin along a (100) direction, whereas the top of the valence band occurs at k = 0 (Fig. 8). There is as yet no experimental confirmation of the energy band structure in diamond. b;l'It also appears likely that the energy band structures in PbS, PbSe, and PbTe are more complicated than the two band model. Thus theoretical calculations of the electronic band structure of PbS by Bell et al.

2akr-2ak,

FIG.9. Schematic diagram of valence and conduction bands in PbS along the 1110 direction [D. G . Bell, D. M. Hum, L. Pincherle, D. W. Sciama, and P. M. Woodward Proc. Roy. SOC.A217, 71 (1953)l.

(21) indicate that the band edges occur a t values away from the origin (Fig. 9). The bottom of the conduction band apparently occurs at a Brillouin zone, but the exact position was not determined. The top of the valence band appears to occur along (110) directions. The minimum vertical separation is about 1.3 ev, whereas the minimum separation between the bands is less than 0.3 ev. These results are in accord with the optical constant data (22) for PbS (Fig. 10) which show a rapid rise in absorption at about 1.3 ev, which may be attributed to vertical transitions; as well as a long wavelength tail extending to 0.3 ev which may be attributed to nonvertical transitions. Similar calculations are not available for PbSe and PbTe. The optical constant data for these materials

16

ELIAS BURSTEIN AND PAUL H. EGLI

however (Fig. 11) indicate that they too have energy band structures in which the band edges do not occur a t the same k value. On the basis of early experimental data the M"'-NV compound semiconductors appeared t o have simple energy band structures. Thus Pearson and Tanenbaum (23) obtained room temperature magnetoresistance data for n- and p-type InSb and p-type GaSb which indicated that the valence and conduction bands have approximately spherical

RY

10 4

-' 3

195"C.Mvsry)& 192'C.(Gibson) 295*C.U\very) &303"C.lGibson)

'

2

1 1 1 I I

1

I

0.5 Energy, ev

I

0.3 0.2

1

FIQ. 10. Intrinsic optical absorption spectrum of PbS. The transmission curves are due to A. F. Gibson, Proc. Phys. SOC.B65,378 (1952) and the reflection curves are due to D. G. Avery, Proc. Phys. SOC.B67, 2 (1954).

energy surfaces. More recent Hall and magnetoresistance data obtained for InSb at low temperatures (24) indicate a more complicated structure for the valence band with maxima a t k # O.* This is also confirmed by an analysis of the absorption edge in InSb by Blount et al. (26) which shows that the absorption close t o the edge involves nonvertical transitions. According to Slater and Koster (26) the structure of the valence bands in the MIIr-NVcompounds should be similar to that of silicon and germanium. They suggest, however, that the lowest conduction band in * The recent cyclotron resonance experiments carried out for InSb by Dresselhaus et aZ.(l*)indicate that the conduction band minimum does occur a t k = 0.

PHYSICS OF SEMICONDUCTOR MATERIALS

17

InSb differs from that in the diamond structure in that it is s-type rather than p-type. Kittel and Herman have recently pointed out that the compounds are actually more complicated valence bands in the MII1-NV than in the diamond structure as a result of the asymmetry in the zinc blende structure (2'7).

3

-a

N 0 c

P 9

Fro. 11. Intrinsic absorptionspectra for PbS, PbSe and PbTe obtained from reflection data [D. G . Avery, Proc. Phys. SOC.B67, 2 (1954)l.

.-,'

0.60 0.40

w

0.20 0.40

11001

(1101

0.60

0.80 -0.5

0

lkla

-

0.5

-0.53

0

0.53

lkla -+

Momentum

Fro. 12. Energy-momentum curves for LiF along [loo] Ill01 and [ l l l ] directions [D. H. Ewing and F. Seitz, Phys. Reo. 60, 760 (1936)l.

The alkali halides and other ionic crystals in which the full band is derived from p levels also appear to have complicated energy band structures. Thus theoretical calculation of the energy band structure for NaCl (2'8), LiF and LiH (as), and BaO (SO) indicated that the p-like valence band is largely degenerate at k = 0 but that one band curves upward, while the other two curve downward from k = 0 (Fig. 12). Seitz (31) has proposed a similar structure for the silver halides in order to account for

18

ELIAS BURSTEIN AND PAUL H. EOLI

their optical properties. Krumhansl (32)has pointed out further that the band structures are probably even more complex than indicated in Fig. 12, possibly showing reversals at points between k = 0 and the end points on the faces of the Brillouin zone as in silicon and germanium. 111. TRENDS IN FUNDAMENTAL PROPERTIES Sufficient information is now available about the semiconductor properties of a wide variety of materials to enable us to understand qualitatively the relations between the semiconductor properties and chemical binding and to indicate trends in the fundamental properties of semiconductor materials. In discussing these relations and trends, however, TABLE 111. Energy Gaps and Belated Properties of Melting point ("C) PbS PbSe PbTe

E d Eo (300" K) (ev) (ev)

1110 -0.37 1065 904 -

0.30 0.22 0.27

(ev/"K) f4 X f4 X $4 X

M1 V - B - N V L - B

Heat of formation Compressibility (kcal/mol) ( X lo7 cm2/kg)

so

loe4

17.5 20.5 30.0

TABLE IV. Energy Gaps and Related Properties of Melting point ("0 AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb

1060

1240 720 1070 940 535

EGO (ev) 1.6 0.80

MIII-NV

0.47 0.27

Compounds

(z)

eo

(ev/"K)

-3 -2.2 2.4 1.1 0.7 1.25 0.35 0.18

18.4 20.7 25.6

22.5 18 17.5

EO (300" K) (ev)

1.5

Compounds

3 5.5

x 10-4 x 10-4

3 . 5 X lo-* 4 X 3 x 10-4

eo2Eo

(ev)

_ 10.1 8.4 11.1 14.0 10.8 11.7 15.9

175 169 180 150 135 48 45

TABLE V. Energy Gaps of M Z I I - N Compounds ~~ ~

Melting point ("C)

EGO (ev)

Eo (ev)

Heat of formation (kcal/mol)

1102 1150 770 550

0.77 0.74 0.36

-

0.8 (300" K) 0.33 (5' K)

19.5 18.3 18.3 12.6

-

-

19

PHYSICS OF SEMICONDUCTOR MATERIALS

TABLE VI. Properties of Isoelectronic Series of Compounds Series A Melting point ("C) Energy gap (ev) Electron mobility (cm2/volt-sec) Nearest neighbor distance (A) Series B Melting point ("C) Energy gap (ev) Electron mobility (cm*/volt-sec) Nearest neighbor distance (A) Compressibility ( X 107 cml/kg)

CuBr

ZnSe

GaAs

Ge

498 5 2.46

(1238) 2.8 2.45

1280 1.1 4000 2.44

958 0.68 3800 2.44

AgI

CdTe

InSb

Sn

(d552) 2.8 30 2.80 40

1045 1.5 300 2.80 23.3

523 0.18 80,000 2.79 -

0.08 1560 2.80 19

TABLE VII. Energy Gaps of Isoelectronic Compounds

Crystal

Melting point ("0

Si Alp Ge GaAs AlSb InP Sn InSb

Interatomic distance

E (300" K) (ev)

1420

(A)

2.35 2.36 2.44 2.44 2.62 2.54 2.80 2.80

1.1

-

-3

0.62 1.1 1.5 1.25 -0.08 0.18

936 1240 1060 1070 525

TABLE VIII. Polar Character of Compound Semiconductors Crystal CUCl CuBr ZnS CdS Sic AgCl PbS

Structure

6,

ZnS ZnS ZnS ZnO ZnO NaCl NaCl

10 8 8.3 11.6 -10.5 12.3 (-70)

6.

3.57 4.08 5.07 5.9 5.9 4.04 17.5

q*/e 1.1 1.0 0.96 -1.3 0.8 -0.8

XI (microns)

53 57 33 -12 -97 -80

6.60

0.18 0.11 0.08 0.08 -0.07 0.17 -0.04

20

ELIAS BURSTEIN AND PAUL H. EQLI

we will restrict our attention for the most part t o homologous series of semiconductors with simple structures. These include (a) the group IV-B elements with diamond structures, (b) the M 1 I I - B - N V - B ) M I I - B - N V I - B , and M 1 - B - N v 1 1 - B compounds having the zinc blende or wurtzite structures, (c) the M I V - B - N V I - B , M I I I - B - N V 1 - B and the M I - B - N V I 1 - B compounds having the sodium chloride or cesium chloride structures, and (d) the M 2 1 1 - * - N I V - B compounds having the fluorite structure. The forbidden energy gaps and mobilities of materials in the various homologous series of semiconductors are summarized together with other pertinent physical properties in Tables 111-VIII. 1. Energy Gap

We note that, with the exception of the PbS group of semiconductors, the forbidden energy gap in the various homologous series of semiconductors decreases with increasing atomic number of the component elements, as would be expected from the fact that the binding energy of the valence electrons decreases with increasing atomic number. I n the PbS group of semiconductors (Table 111), PbSe is found t o have a smaller energy gap than PbTe. This anomaly in the energy gap trends may be attributed t o the complex structure of the bands and t o the fact that the forbidden energy gaps of PbS, PbSe, and PbTe do not differ appreciably from one another. There is apparently a fortuitous balancing of factors which control the position of the band edges but which do not affect other intrinsic properties. Thus, the optical dielectric constants and the major absorption peaks in the intrinsic absorption spectra (Fig. 11) do follow the correct order, as do the other major physical properties of these materials (Table 111). It is of interest to note that most semiconductors exhibit a negative temperature dependence of the energy gap having about the same value, i.e.,

aE/aT

= -3 to -5

X

ev/deg.*

The PbS group of semiconductors, on the other hand, exhibits a positive ev/deg.t The small varitemperature dependence of about +4 X ation in energy gap within a homologous series of compounds in which one component is constant appears to be a characteristic of ionic compounds. The alkali metal chlorides, for example, show only a small vari-

* Tellurium exhibits a much smaller negative temperature dependence, e.g., = -1 X ev/deg. This is due to almost equal but opposite contributions

aE/aT

from the change in lattice constant and to pure temperature effects. t Recent measurements of Macfarlane and co-workers (private communication) indicate that MgZSn also exhibits a positive temperature dependence of the optical energy gap.

PHYSICS O F SEMICONDUCTOR MATERIALS

21

ation in their optical energy gap. I n the PbS group of semiconductors, it may be attributed to the fact that the character of both the conduction and valence bands is determined largely by Pb ions. We may expect a larger difference in energy gap, however, between the P b compounds and the corresponding Sn compounds having sodium chloride structures, e.g., between PbTe and SnTe. The variation in energy gap among members of a homologous series is appreciably larger in the covalent semiconductors. This is shown, for example, by the data for In(P,As,Sb) as the V-B elements are changed and for (Al,Ga,In)Sb as the 111-B elements are changed (Table IV). A large variation is to be expected because the energy bands in covalent crystals are made up of a combination of the energy levels of the two components. The MgzSi group of semiconductors also exhibits an appreciable variation in energy gap (Table V). The type of binding in these materials is not too well established. They presumably have an appreciable ionic character, since the component elements have an appreciable difference in electronegativity. The character of the end member of this group, MgZPb, has not been finally determined. It has the same structure as the other Mzll-N1v compounds in the group and exhibits a high heat of formation, so that we may reasonably expect it to be a semiconductor having a small energy gap, although experiments have shown only metallic character. * The influence of ionic and covalent character is illustrated by the data for two isoelectronic series of compounds having the wurtzite or zinc blende and diamond structures (a) CuBr, ZnSe, GaAs, Ge and (b) AgI, CdTe, InSb, Sn (Table VI) (34). There is no appreciable variation of the lattice constants among the compounds in each isoelectronic series. In going from the ionic MI-NV*I compounds to the covalent group IV-B elements, the energy gap decreases continuously, while the melting points increase initially and then decrease. Thus, the admixture of covalent binding decreases the energy gap of the ionic crystals and the admixture of ionic binding increases the energy gap of covalent crystals. The interpretation of the effect of the ionic and covalent character on the melting points of the compounds in the isoelectronic series is more involved since the melting point is determined by the free energy of the liquid as well as that of the crystal. Melts of the ionic compounds retain their ionic character whereas covalent compounds on melting invariably become metallic, t

* The data reported by Robertson and Uhlig (33)are based on room temperature measurements on relatively impure specimens. That these specimens exhibited metallic behavior can reasonably be attributed both to a high concentration of charge carriers and to a small energy gap. t This has been demonstrated, for example, in Ge (36)and InSb (36).

22

ELIAS BURSTEIN AND PAUL H. EOLI

i.e., the covalent bonding is destroyed. The initial increase in melting point, in going from the MI-NvII compounds t o the group IV-B elements, is attributed to the fact that there is initially a n increase in the ionic charge of the components. The subsequent decrease in melting point is similarly due in part to the fact that the ionic charge and therefore the coulombic contribution to the binding energy decreases t o zero as the Compounds become more homopolar. In the highly ionic crystals, the cohesive energy and therefore the melting point is determined primarily by the coulombic forces between the ions, and there is no appreciable contribution as such from the electronic energy of the valence electron. There is, consequently, no relation between the cohesive energy and the energy gap even within a given homologous series except perhaps in a very indirect way. An extreme example of the absence of any relation between cohesive energy and energy gap is the solid rare gases, which have relatively large energy gaps and yet are characterized by very low melting points. I n the highly covalent crystals, on the other hand, the cohesive energies of the crystal and therefore the melting point is determined by the electronic energy of the valence electrons in the covalent bonds, and we may expect a more direct relationship between the cohesive energy and the energy gap. The MIII-NV compounds made up of elements in the same row of the periodic table, e.g., GaAs and InSb, have larger energy gaps and higher melting points than their isoelectronic components among the group I V elements. This may be due to the admixture of ionic bonding in the MIII-Nv compounds (Table VII). I n the case of GaAs, the relative increase in energy gap is larger than the relative increase in melting point so that it actually has a lower melting point and yet a larger energy gap than silicon. The effect is particularly large for MIII-NV compounds made up of elements from different rows of the periodic table. These compounds presumably have an even greater amount of polar character, due t o larger differences in electronegativity, than the compounds made up of elements from the same row. Thus, I n P and AlSb have larger energy gaps and smaller melting points than silicon. On the other hand, InAs has a much smaller energy gap than Ge and yet has a somewhat larger melting point. Goodman (37) has recently suggested that the energy gaps of compounds with ZnS structure can be predicted on the assumption that (a) the energy gap increases with melting point, (b) compounds with the same melting point and the same difference in electronegativities of the components will have the same energy gaps, and (c) an increase in electronegativity difference will increase the energy gap. This procedure places too much reliance on the electronegativity of the elements, which is not a rigorously defined property. Using this procedure, for example, Goodman predicted

PHYSICS O F SEMICONDUCTOR MATERIALS

23

that InAs would have an energy gap in the range of 0.8 to 1.0 ev, whereas InAs has been found to have an energy gap of 0.35 ev. Data for the various homologous series show that the optical dielectric constants increase with decreasing energy gap. Moss (38)has pointed out that eo2& was approximately constant for the group IV-B elements and for silicon carbide. As shown by Briggs et al. (39) this relation also holds fairly well for the MIIr-NV compounds except for InAs and InSb (Table IV). We note that in the compound semiconductors for which both optical and thermal data are available (i.e., for semiconductors with energy gaps less than 2 ev), the optical and thermal energy gaps are in fairly good agreement. This agreement between the optical and thermal energy gaps is attributed to the fact that the optical dielectric constants in these materials are fairly large, so that the factor (e, - eo)/c,eo, which determines the difference between optical and thermal energy gaps in polar crystals, is small. For ionic crystals with simple cubic structures, the static dielectric constant is related to the optical dielectric constant and the transverse optical lattice vibration frequency according to the expression (40)

where q* is the “dynamic” effective charge of the ions,* u J 2 ~is the transverse optical lattice vibration frequency, N is the number of “molecules” per unit volume, and M is the reduced mass of the ions. The value of e, - e, decreases with decreasing effective charge q* of the ions, but in2)/3Ia.Accordcreases rapidly with increasing co due to the factor [(e, ingly, in the appreciably ionic semiconductors, in which q* remains appreciable, (c, - e,) increases as co increases with decreasing energy gap. (Q- e,) decreases however with increasing homopolar character (i.e., as the effective charge decreases) and goes t o zero for the group IV elements. Information about G , G,, and at is of particular value in establishing the polar character of single crystals. Unfortunately, the determination of es is somewhat difficult in the case of semiconductors because of the large contribution from the free carriers to the real and imaginary parts of the dielectric constant. This difficulty may be avoided by making measurements at low temperature where the carriers are bound to impurities (41).Information about ea can also be obtained indirectly from a combination of infrared lattice vibration absorption and compressibility

+

* The magnitude of q* includes contribution from the change in effective charge during lattice vibration, e.g. charge deformation, as well as the static charge on the atoms.

24

ELIAS BURSTEIN AND PAUL H. EGLI

data by means of the expression (42)

where K is the compressibility, and R is the nearest neighbor distance. The known values of e0, e,, and q* for various polar semiconductors are given in Table VIII. Information about the effective charge of the atoms can also be obtained from the strength of the infrared lattice vibration absorption bands. Thus the group IV elements which have zero effective charge do not exhibit any fundamental optical lattice vibration absorption (43).

$100-

+

-

Sic

ZnS

5 80L

W

L

-

60-

z w

-

40w -

20

-

6

7 8 9 10

I 15

I 20

I

30

I 40

I 50

I

WAVE LENGTH (MICRONS)

I l l 1 100

I

200

FIG.13. Infrared (lattice vibration) reflection spectra of compound semiconductors; Sic and ZnS [G. Picus (unpublished)]; InSb [H. Yoshinaga, Molecular Spectroscopy Symposium, Ohio State University, 14551; PbS [J. Strong, Phys. Rev. 58, 1818 (1931)l.

The compound semiconductors such as S i c and InSb do however exhibit appreciable fundamental optical lattice vibration absorption bands (Fig. 13). The band exhibited by InSb is considerably narrower than those exhibited by the more ionic compounds such as ZnS and PbS, indicating a low oscillator strength and a small effectivecharge on the atoms. This is associated with the small difference in electronegativity of indium and antimony. Sic, on the other hand, appears to have an appreciable polar character (44) (Table VIII) in keeping with the large difference in electronegativity of silicon and carbon. 2. Mobility

Reliable data on the intrinsic lattice scattering mobilities have been obtained for a relatively small number of semiconductor materials for which relatively pure, perfect, single crystal specimens are available. The available data for the other semiconductor materials are nevertheless useful for indicating lower limits to the intrinsic mobilities.

25

PHYSICS OF SEMICONDUCTOR MATERIALS

The data for the group IV elements (Table IX) indicate that the mobilities of the charge carriers in diamond are higher than those in silicon while the mobilities of gray tin and germanium are comparable in magnitude. Since the values reported for the mobilities of electrons and holes in gray tin are based on relatively poor specimens, they represent only lower limits t o the lattice scattering mobilities. It is therefore possible that the intrinsic mobilities of gray tin may exceed those of germanium. The higher mobilities of electrons and holes in diamond relative to silicon are apparently due t o the very much smaller amplitudes of vibration in diamond associated with the larger Debye temperature. When the mobilities are compared a t the same reduced temperatures, TABLE IX. Mobilities of Group IV Elements fin (300"K) (cm2/volt-sec)

Diamond Silicon Germanium Gray tin

N1800

1200

3900 -2000 (273" K)

pp (300' K) (cm2/volt sec)

1200 500 1900 -1000 (273" K)

Debye temperature (OK) 1950 ti58 362 260

we find that the mobilities actually increase in the order of increasing atomic number of the elements. The interpretation of the trend in mobility among the group IV-B elements is complicated by the existence of other charge carrier scattering processes which are presumably responsible for the deviations from the T-9%temperature dependence normally expected for nonpolar scattering in semiconductors with simple energy bands. Prior t o the semiconductor investigation of the MIII-NV compound semiconductors it was generally assumed that the mobilities would increase uniformly with decrease in polar character and that the group IV elements with diamond structure would have the highest mobilities. It was therefore particularly interesting when Welker (46) reported that some of the compounds exhibited electron mobilities which were higher than those of the group IV elements. The higher mobility of the MIII-NV compounds was a t first attributed to a decrease in the amplitude of lattice vibrations resulting from the presence of a small amount of ionic character which was assumed to be insufficient to introduce any appreciable polar scattering. It is now believed that, although there may be some effect due t o reduced lattice vibration amplitudes, the higher mobilities of the M111-N~compounds are associated with lower effective masses of the carriers. Thus, InAs and InSb, which have much larger electron mo-

26

ELIAS BURSTEIN AND PAUL H. EGLI

bilities than the group IV elements, have much smaller effective masses (Table X). The difference in the energy band structure of the two groups of materials must also be taken into account when comparing their mobilities. The differences in properties of the MIII-NV compounds and group IV elements is thus associated with differences in the nature of their energy bands, i.e., difference in effective masses and type. of energy band structure. It is very likely that these differences arise from degeneracy effects TABLE X. Mobilities of

AlSb GaAs GaSb InP InAs InSb

MI*I-NV

pn (300’ K) (cm4/volt-sec)

p, (300” K) (cm*/volt-sec)

1200 4000 4000 3400 30,000 77,000

200 200 850 650 200 -1250

Compounds

(?)”*

>

-

m

-

-

0.28 -

0.1 0.083

* Density of states mass. t Effective masses calculated from (m,mp/n)” and

spherical energy surfaces. 3 See text for values obtained by other methods. From thermoelectric power data.

m,t

(0.03)%0

-0.2 0.07 0.03%

(pn/pp)

mpt m

-0.3 0.3 0.2

on assumption of

in the centrosymmetric structure of the group IV elements which are not present in the asymmetrical zinc blende structure of the M111-N~compounds whose sub-lattices are not identical. Thus the effect of a small amount of ionic character is to make the two sub-lattices nonidentical and thereby to remove the degeneracy which exists in the group IV elements. High mobilities in compound semiconductors are not limited to the MIII-NV compounds. HgSe ( p n > 10,000 cm2/volt-sec) and HgTe ( p n = 17,000 cm2/volt-sec) also have higher mobilities than those of the group I V elements (37), which may be attributed to small effective masses and to the absence of any appreciable polar scattering. As shown by the available data for the MIII-NV compounds there is an increase in electron mobilities with increase in atomic number associated with a corresponding decrease in effective mass. The corresponding increase in hole mobility is considerably smaller, so that the ratio of electron to hole mobility increases rapidly. The MII-NVI compounds-ZnO, ZnS, CdS, and CdSe-exhibit only n-type character. Jenny and Bube (46)suggest that this is due t o the

27

PHYSICS OF SEMICONDUCTOR MATERIALS

presence of very deep traps having large capture cross sections for holes rather than to low hole mobilities. Their data indicate that MII-NvI compounds made up of high atomic number elements in which the hole traps are not as deep do exhibit both n- and p-type character. Wlerick (4?’), however, suggests that the absence of p-type character in the low atomic number compounds is associated with low mobilities arising from the large effective masses of the holes. The variation of mobility with variation of ionic and homopolar character may be seen from the electron mobility data for the isoelectronic series of compounds AgI, CdTe, InSb, and Sn (Table VI). InSb has the highest mobility, and the mobility of Sn is greater than that of CdTe and AgI. Seraphin (48) has recently carried out an analysis of a simple one-dimensional Kronig type model for the ZnS structure with varying character which appears to account for the high mobility ratio of the III-V compounds and for the trend in properties of the isoelectronic series of compounds. He finds that (a) the effective mass of the holes increases monotonically with increasing ionic character, (b) the effective mass of electrons decreases at first to a minimum value and then increases with increasing ionic character, and (c) the forbidden energy gap increases with increasing ionic character. Actually the decrease in mobilities in going from InSb t o CdTe to AgI is due to increasing contribution from polar lattice scattering resulting from an increasing ionic character as well as to an increase in effective mass. The perturbation theory of polar lattice scattering yields an expression for the mobility of the form (49)

’

-

cm2/volt-sec, T

<< 0

(6)

where 8/T = hwl/kT; w1/2r is the frequency of longitudinal optical lattice vibration; and the temperature dependence is given by the exponential factor. A similar expression is also obtained from polaron scattering theory (60). The factor eoea/(eo - E,) is small for ionic crystals such as the alkali and silver halides where (e8 - 6,) is appreciable. It is very large for homopolar semiconductors such as the M1ll-Nvcompounds dueboth to small values of (e, - e,) and to large values of c,E,. There are additional lattice scattering mechanisms which may play a role in polar crystals. Thus, Meijer and Polder (61) have shown that, in piezoelectric crystals, the acoustic modes play a role in “polar lattice scattering” and yield a mobility which has a T-w temperature dependence. They also point out that, in ionic crystals with NaCl structure, the difference in the mass of the ions can in principle also be responsible for polar scattering for which the mobility would have a T-% temperature dependence.

28

ELIAS BURSTEIN AND PAUL H . EGLI

Typical mobilities and related properties of polar crystals are summarized in Table XI. The mobilities of electrons in the alkali halides, which are approximately one or two orders of magnitude smaller than those in the group IV elements, are found to be in reasonable agreement with theory (52). On the other hand, AgCl appears to have a T-3* mobility temperature dependence (63) rather than an exponential temperature dependence which would be expected for polar scattering. In CdS, Kroger et al. (64) are able to account reasonably well for the temperature dependence of electron lattice scattering mobilities over a range of temperatures from 100°K to 700°K by a combination of polar and nonpolar TABLE XI. Mobilities in Polar Materials p, (300' K) pn (-90" K ) (cmz/volt-sec) (cmz/volt-sec)

NaCl KCl KBr KI AgCl CdS PbS

25

-

12.4 50 210 580

250 100 110 155 300 1500 10,000

fado

e.

2.2 211 2.3 2.7 4.0 5.9. 17.5

€.

5.7 4.7 4.8 4.9 12.3 11.6

(-70)

(€a

-C

J

3.7 3.7 4.4 5.9 5.9 12 23

scattering. From a comparison of theory and experiment they arrive a t an electron effective mass of 0.2 to 0.3m. The relatively high mobility of electrons and holes in the PbS group of semiconductors, which are appreciably ionic, is due in part to the relatively large values of e, and eoe,. The mobilities of the carriers have a T-$$temperature dependence (55). Petritz and Scanlon (66) have calculated the theoretical mobility for PbS based on a combination of polar and nonpolar lattice scattering mobilities. They get a reasonable fit in magnitude with the data a t low and intermediate temperatures.* They are, however, not able to account for the T-54 temperature dependence of the mobilities. The mobility of free carriers in the transition metal oxides is quite small (67). In these materials, the low mobility is very likely due to relatively large effective masses which are associated with the very narrow d bands.

* Their calculations are based on a value of 17.5 for the static dielectric constant obtained from the International Critical Tables. This is only slightly larger than the value for the optical dielectric constant and is clearly much too low in view of the polar character of PbS. Estimates from the compressibility and the residual ray frequency indicate a value of the order of 70.

PHYSICS OF SEMICONDUCTOR MATERIALS

29

IV. IMPURITY AND LATTICEDEFECT CENTERS 1. Character of Levels in the Forbidden Band Impurity and lattice defect centers affect the electrical and optical properties in a number of ways. They act as the source of majority charge carriers under thermal, impact, and optical ionization, as radiative or nonradiative trapping centers, * and as recombination centers. The varied Capture processes

Ionization processes

Ionization processes

Capture processes

(b)

FIG. 14. Dual character of energy levels within the forbidden band, (a) donor centers with single energy level; (b) acceptor centers with single energy level. Do and A0 represent the un-ionized states of the donor and acceptor centers; D+ and Arepresent the ionized states of the donor and acceptor centers; ED, and ED,,are the ionization energies of the donor centers as an electron inceptor and as a hole inceptor; Ed, and EA, are the ionization energies of the acceptor center as an electron inceptor and as a hole inceptor.

behavior of the impurity and lattice defect centers is in large part due to the dual nature of their energy levels which arises from the complementary relationship between electrons and holes. Thus a level to which a hole is bound corresponds to an empty level with respect to electrons so that it can behave as a receptor of an electron as well as an inceptor of a hole (Fig. 14). Conversely, a level t o which an electron is bound constitutes an empty level with respect to holes so that it can behave as a hole receptor as well as an electron inceptor. The electron or hole oc-

* The trapping of the free carriers may take place by radiative capture with the emission of photons or by nonradiative capture with either the emission of phonons or the excitation of other free carriers (cf. section V).

30

ELIAS BURSTEIN AND PAUL H. EOLI

cupancy of a level and therefore its behavior as a receptor or as an inceptor depends on its position relative to the Fermi level. In view of the dual nature of the energy levels within the forbidden band, the usual terms donor and acceptor normally applied to impurity and lattice defect centers do not adequately describe their electrical and optical behavior. It is nevertheless still useful to use the term donors for impurity and lattice defect centers which yield electrons when introduced into the lattice, and the term acceptors for those centers which yield holes when introduced into the lattice. Donors are positively charged centers when they are empty with respect to electrons (e.g., when their levels lie well above the Fermi level), while acceptors are negatively charged centers when they are empty with respect to holes (e.g., when their levels lie well below the Fermi level). For deep lying levels, the sign of the Hall constant does not indicate whether the impurity is an acceptor or a donor type center. The character of a deep lying impurity level can be determined, however, from its effect on the charge carrier density when added to n-type and to p-type materials containing appropriate concentrations of shallow donor and acceptor levels, respectively (58). The electrical behavior of impurity and lattice defect centers may be characterized by (a) the position of their energy levels within the forbidden band, (b) the ionization cross section of the center as an electron inceptor, (c) the ionization cross section of the center as a hole inceptor, (d) the capture cross section as an electron receptor, and (e) the capture cross section as a hole receptor. There are three major ionization processes: (1)ionization by phonons, (2) ionization by photons, and (3) impact ionization by free charge carriers. The capture processes are the reverse of these ionization processes and the capture cross sections are related to the corresponding ionization cross sections, by the principle of detailed balance. The cross sections for a particular capture or ionization process will be largest when the ionized center and the free carrier are of opposite charge, intermediate if the center is neutral, and smallest when the charges are of the same sign. This results from the fact that for any particular perturbation the matrix element that enters into the calculation of the cross sections will depend on the degree of overlap between the wave functions describing the bound state and the free state of the charge carrier. The wave function of the bound state will, of course, be localized about the impurity center, and the amplitude of the free state wave function in the vicinity of the impurity will depend on the character of the Coulomb forces between the ionized center and the free carrier. If there is an attractive force between the center and the free carrier-

PHYSICS O F SEMICONDUCTOR MATERIALS

31

as in the case of ionized donors and free electrons-the amplitude of the free carrier wave function is increased in the region of the impurity and the interaction matrix element is large. Conversely, if there is a repulsive force, the amplitude of the free carrier wave function is decreased and the matrix element is small. If the ionized center is neutral, the free charge carriers will be represented by plane wave functions whose amplitude at the center is the same as elsewhere in the material. The matrix element will therefore be intermediate. The ionized state of a donor with respect t o electrons is charged positively, but when considered as the source of a hole the ionized state is neutral. Consequently the ionization cross sections of a donor with respect to electrons is larger than the cross section with respect to holes. The capture cross section of the ionized donor with respect t o electrons is also larger than the capture cross section of the neutral center for holes. Correspondingly, the acceptor centers will have larger cross sections for ionization and capture of holes than for electrons. I n addition t o these effects the relative magnitudes of the ionization and capture cross sections of a particular center for holes or electrons will depend on the position of the corresponding energy level in the forbidden band. The closer a level is t o a particular band, the larger become the cross sections for processes involving the carriers associated with that band. There will be a particularly pronounced difference in cross section for electrons and holes for shallow donor and shallow acceptor levels. Thus, in silicon and germanium the group I11 acceptors behave primarily as hole receptors and inceptors, and the group V donors behave primarily as electron receptors and inceptors. The behavior of impurity and defect centers in semiconductors varies in Eiome details with the type of chemical binding. Among the homopolar semiconductors, extensive studies have been made of a large number of impurities and defect levels in both silicon and germanium; and isolated pieces of information are available for the M I I I - N V compounds. For the ionic compounds much of our knowledge is derived from the extensive studies of alkali and silver halides. Additional information is available from studies of luminescence of the group 11-B sulfides and from recent studies of the semiconductor and photoconductor properties of the PbS and CdS groups of semiconductors. 2. Homopolar Semiconductors

Information is now available on the behavior of a wide variety of impurity elements in silicon and germanium (69). We may expect qualitatively the same behavior in other homopolar semiconductors including the M I I I - N V compounds.

32

ELIAS BURSTEIN AND PAUL H. EGLI

I n silicon, germanium, gray tin, and presumably also in diamond, group 111-B elements, such as B, Al, Ga, and In, which have one less valence electron than the atoms of the host lattice, behave as acceptors. Group V-B elements such as P, As, and Sb, which have one more valence electron, behave as donors. The early ionization energy data for these elements in germanium indicated that the two groups of elements had essentially the same ionization energies (Er = 0.01 ev) (60). The data for silicon indicated roughly the same ionization energy for boron acceptor centers (EI = 0.08 ev) as for phosphorus donor centers (EI = 0.06 ev) TABLE XII. Impurity Ionization Energies in Silicon*

B

A1 Ga In P AS

Sb Li Au

0.044 (D) 0.049 (D) 0.039 (D) 0.033 (D) 0 . 3 (1)

0.045 ( A ) 0.057 ( A ) 0.065 ( A ) 0.16 ( A )

0 . 3 9 (?)

(D)indicates donor type level and (A) acceptor type level.

* After J. Burton, Physica 20, 845

(1954).

t Values obtained from measurements on n-type materials. 1Values obtained from measurements on p-type materials. (61). The low ionization energies of the group I11 and group V elements in silicon and germanium were explained in terms of a hydrogen model for the un-ionized impurity centers (62).I n this model the group I11 and group V elements, which enter into covalent bonding with the atoms of the host lattice, are pictured as substitutional atoms with one extra or one less nuclear charge. The charge carriers are bound to the impurity atom by a Coulomb potential which is reduced by the dielectric constant of the medium (eo = 11.5 for silicon and 16 for germanium), so that the and ) the orbit of the ionization energy is given by Er = ( E ~ / e , ~ ) ( m * / m ground state is given by a = aoe,(m/m*) where EH and a0 are the ionization energy and Bohr orbit of hydrogen and m is the rest mass of the electron. The simple hydrogen model for the impurity centers has also been derived more rigorously, expressing tlhe wave functions of the bound charge carrier as a linear combination of Wannier functions (63). For high dielectric constants which spread the Coulomb potential, one

PHYSICS OF SEMICONDUCTOR MATERIALS

33

obtains a differential equation which for spherical energy surfaces is essentially the Schrodinger equation for a hydrogen-like center with a potential given by V ( r ) = - e e 2 / w and an effective mass m* in place of the real mass. Recent data indicate that there are small but reproducible differences in ionization energy among the group I11 and group V elements in germanium (64) (Table XII) and even larger variations in ionization energy TABLE XIII. Impurity Ionization Energies in Germanium * Element B A1 Ga In P As Sb Li

Ec - Ert

0.0120 (D) 0.0127 (D) 0.0096 (D) 0 . 0 1 (D)

Zn cu Au

0.20

(A)

Ni Fe co Pt Mn

0.30

(A) (A) (A) (A) (A)

0.27

0.31 0.20 0.35

Er - E,$ 0.0104 ( A ) 0.0102 ( A ) 0.0108 ( A ) 0.0112 ( A )

0.095 0.029 0.04 0.30 0.05 0.15 0.22 0.34 0.25 0.04 0.16

(A) (A) (A) (A)

(D)

(A) (A) (A) (A) (A) (A)

(D)indicates donor type level and ( A ) acceptor type level. *After J. Burton, Physica 20, 845 (1954) including recent data reported by W. W. Tyler and co-workers. t Values obtained from measurements on n-type materials. $ Values obtained from measurements on p-type materials.

among the group I11 elements and among the group V elements in silicon (65) (Table XIII) which are not expected on the basis of a simple hydrogen model.* The value for indium in silicon is especially large compared to the other group I11 impurities in silicon. Low temperature optical studies have demonstrated optical absorption and photoconductivity due to un-ionized group I11 donor and group * Differences between recent values for the ionization energy of boron and phos-

phorus in silicon and the earlier values reported by Pearson and Bardeen (61) are due to peculiarities arising either from nonuniformity or from unknown impurities in Pearson and Bardeen’s samples.

34

ELIAS BURSTEIN AND PAUL H. EGLI

V acceptor impurities in both germanium and silicon (66),and have also revealed the existence of excited states of the impurity centers in silicon (67).* The optical processes which may occur for donor and acceptor centers include (a) photoionization absorption involving transitions from the ground state of the donor center to the conduction band, and (b) optical excitation absorption involving transitions from the ground state to excited states of the donor center, which should appear as narrow bands

FIG. 15. Schematic energy-momentum diagram of bound states due to shallow donor and acceptor centers in silicon and germanium and the possible optical transition of the bound carriers.

beyond the long wavelength photoionization absorption limit. For acceptors, the photoionization absorption should also involve transitions from the ground state of the acceptor to the U S valence band as well as t o the 2rl and v 2 valence bands (Fig. 15). Fairly complete optical data are now available for the group I11 acceptor impurities, B, All Gal and In, in silicon (68). It is found that the variations in ionization energy among the group I11 elements in silicon are accompanied by variations in the character of the excitation and photoionization absorption spectra (Fig. 16). For boron, the positions of the excitation bands correspond closely to the 1s - 2p, 1s - 3 p and 1s - 4p transitions of a hydrogen-like center although the oscillator strengths of the bands differ from those for a hydrogen model. The optical ionization energies of boron can therefore be calculated fairly accurately from the positions of the excitation bands and are found to be in close agreement with the thermal value. The positions of the excitation bands *The measurements, which have thus far been limited to wavelengths below 38 microns, do not extend far enough into the infrared to yield information about the excited states of the group I11 and group V impurity centers in germanium.

35

PHYSICS OF SEMICONDUCTOR MATERIALS

due to aluminum, gallium, and indium, unlike those of boron, do not correspond to a hydrogen-like model. As in the case of boron, the oscillator strengths of the excitation bands deviate appreciably from the corresponding oscillator strengths of a hydrogen-like model. The deviations so

-

BORON

-

40

5

ALUMINUM

46

0.1 O t

-I 6 0

8 v)

-

o.8

-

a6-

p

0.4

I

I 0.1

I.2-

l.O"IO'6

B z

I

I 0

I

I

I

I

PnoioN ENERGY tw)

I

I

0.2

I

I

I 0.3

INDIUM

-

&

p a

Q2-

I

, + I .,

,

FIQ.16. Absorption spectra due to un-ionized group I11 acceptors in silicon. For boron-doped silicon the dashed line is the theoretical photoionization absorption spectrum for a hydrogen model.

become more pronounced in going from boron to aluminum t o gallium t o indium. The shape of the photoionization absorption spectra also shows a trend away from a hydrogen-like model. Excitation bands have also been investigated in n-type silicon (69).The data are as yet incomplete

36

ELIAS BURSTEIN AND PAUL H. EGLI

and the assignment of the excitation bands to specific impurities is still tentative. It is obvious from the variation among the properties of the group I11 and group V elements that a more detailed model is needed which takes into account the nature of the impurity atom and its interaction with neighboring atoms of the host lattice as well as the structure of the conduction and valence bands. Koster and Slater (70) have pointed out that, in treating the impurity level on the basis of hydrogen wave functions, the dielectric constant of the medium does not affect the potential of the impurity atom itself, although it does shield the impurity atom’s field a t the position of the neighboring atoms. They suggest that a more appropriate model for the impurity center would be one with a Coulomb potential at the neighboring atoms reduced by the dielectric constant plus a potential well at the central atom. The wave function for such a model would be more concentrated at the impurity center than it would be for the simple hydrogen model, and the ionization energy would be larger. They also note that when the potential well is not too deep, which is presumably the case for the group I11 and group V elements in silicon and germanium, its contribution to the wave function in media with high dielectric constants will not be large. The effective potential at the impurity atom, in the case of the group I11 and group V elements in silicon and germanium, will depend on the size of the impurity atom and on the nature of the ion core. It may be expected to depend on the extent to which the tetrahedral covalent bonds are deformed by differences in the tetrahedral radii of the impurity and host atoms, as well as on the strength and polarity of the covalent bonds between the impurity atom and the neighboring atoms of the host lattice. The smaller variation of the ionization energies among the group I11 and group V elements in germanium than in silicon may be attributed to the fact that the effect of the potential well at the impurity center will be smaller, the larger the dielectric constant. The relatively large ionization energy of indium is probably associated with its relatively large tetrahedral radius compared to silicon and the consequent appreciable deformation of the covalent bonds between the impurity atoms and the neighboring silicon atoms. Various attempts have been made t o develop a more exact theory for the ionization energy of the group I11 and group V elements in germanium and silicon by taking into account the structure of the energy bands (7i). As a result of the degeneracy of the valence band, the effective mass theory for acceptors yields in place of the ordinary Wannier equations a set of coupled differential equations which has not yet been solved. For donors, the effective mass theory leads to a Schrodinger-like equation with different masses for motion in the three coordinate directions for

PHYSICS OF SEMICONDUCTOR MATERIALS

37

which it is possible to obtain a good approximate solution. However the theoretical values for the ionization energies are found to be smaller than the experimental values. This is attributed to effects at the central atom which cannot be taken adequately into account. Kohn and Luttinger, and Kleiner (72) have also derived the position of the excited levels of the donor centers in silicon. Kohn (73) has moreover calculated the oscillator strength for transitions from the ground state to excited p-like states and has suggested an interpretation of the observed excitation bands due to the low ionization energy donor impurities. This interpretation yields good agreement between theory and experiment. Although the effective mass theory does not yield information about the ionization energies of the impurities because of central atom effects, it is possible t o obtain this information by combining the experimental energy difference between the excited states and the ground state with the well established theoretical position of the excited p-like states. Elements whose valence differs from that of silicon and germanium by more than one, such as the group I and group I1 elements and the transition elements, generally introduce more than one energy level into the forbidden band (59). These elements tend to have relatively deep lying levels which cannot be readily interpreted in terms of a simple atomic model for the impurity center. Tithium is an exception which behaves as a donor with a single energy level whose ionization energy is essentially equal to those of the group V elements. Accordingly, lithium is believed to be present in the lattice as an interstitial, singly-charged positive ion to which an electron can be bound in hydrogen-like states (59). Although the early data for zinc in germanium obtained by Dunlap (58) indicated a single shallow acceptor level 0.03 ev above the valence band, more recent data obtained by Tyler (74) indicate a second acceptor level 0.095 ev above the valence band. Data for copper, nickel, iron, cobalt, platinum, and manganese in germanium indicate that they introduce two acceptor levels* within the forbidden band (Table XIII). Gold also exhibits two acceptor levels in germanium, one 0.20 ev below the conduction band and one 0.15 ev above the valence band (58).Morton and co-workers (75) have recently demonstrated that gold also introduces a third level a t 0.05 ev above the valence band which they believe to be an acceptor type level. The existence of the third level has been confirmed by Dunlap (76) who, however, presents evidence that the third level is donor type in char-

* For impurity centers which introduce multiple levels, the term “acceptor level’’ is used for those levels for which the benter is negative when the levels are empty of holes, and “donor level” is used for those levels for which the center is positively charged when the levels are empty of electrons.

38

ELIAS BURSTEIN AND PAUL H. EOLI

acter. The available data for gold in silicon indicate that gold exhibits two deep lying energy levels. One of the levels is donor type, but the character of the other level has not yet been established. The data for gold in germanium indicate that the various energy levels due to gold are present in approximately equal concentration and therefore presumably arise from the same center. The two levels due to copper also appear to have approximately equal concentration (77), but the data are not sufficiently accurate to establish definitely whether both bands arise from the same center. Dunlap has proposed a model for gold centers in which the different energy levels are aseociated with successive states of ionization (68). Thus the donor-type level 0.05 ev above the valence band is associated with the ionization states Auo and Au+, i.e. Auo + Au+

donor

+ e-,

(7)

the acceptor level 0.15 ev above the valence band is associated with the ionization states Auo and Au-, i.e. Auo + Au-

acceptor

+ h+,

(8)

and the acceptor level 0.20 ev below the conduction band is associated with the ionization states Au- and Au- i.e. Au-

acceptor

+ Au-

+ h+.

(9)

The impurity centers having two or more energy levels within the forbidden band may be expected to exhibit complex behavior because of the dual electronic character of each of the levels. For impurities such as nickel which have only two acceptor levels, the thermal ionization energies obtained for n- and p-type material in which the Fermi level is appropriately located correspond to the ionization processes

M-

+ Egn+ M- + e-

Mo

+ El=+

(n-type) in which M- behaves as an electron inceptor, and

M-

+ h+

(p-type)

(10) (11)

in which M o behaves as a hole inceptor. Two additional ionization processes are possible* MEl,,--+ M o e(12) * Such multiple ionization processes have been proposed by Morgan (78) to

+

+

account for the spectral dependence of time constants which he observed in his photoconductive studies of germanium at wavelengths just beyond the intrinsic absorption edge.

39

PHYSICS OF SEMICONDUCTOR MATERIALS

and

M-

+ Ez, + M - + h+.

(13)

We note that M- is simultaneously an electron inceptor (12) and a hole inceptor (13) as well as simultaneously a hole receptor and an electron receptor. A consideration of the two cyclic processes

and

Mo M-

+ Eln+ +

M M -o

+ e-h+ where El, + El, +

M - -k Ezn ME2,+ M- + eh+ where Ezn --f

EQ

(14)

+ E2, = EQ

(15)

=

shows that the M- electron inceptor level occurs a t the M o hole inceptor level, and the M- hole inceptor level occurs a t the M- electron inceptor

FIG.17. Energy levels due t o acceptor type impurities which introduce two energy levels within the forbidden band.

level (Fig. 17). The relative importance of the ionization and capture processes will depend on such other things as the position of the Fermi level and the relative magnitudes of the ionization and capture cross sections of M-, M-, and MO. For gold-doped germanium, a second ionization energy may be obtained in p-type material corresponding to the ionization process

M+

+ Esp+

Mo

+ h+

(16)

in which M+ behaves BS a hole inceptor. Here, too, an additional ionization process is possible

Mo

+ Ean+ M+ + e-.

(17)

40

ELIAS BURSTEIN AND PAUL H. EGLI

Thus for gold-doped germanium the M o level, like the M- level, is simultaneously an electron inceptor and a hole inceptor, as well as simultaneously a hole receptor and an electron receptor. Except for zinc- and copper-doped germanium, the solubility of the impurities with deep-lying levels is too small to allow direct measurement of their optical properties by absorption measurements. Information about the optical absorption of the low-solubility impurities can, however, be obtained from impurity photoconductivity studies. Thus Newman and his co-workers (79) have carried out an extensive investigation

10-1

-

10-210-3

-

10-4

-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 Photon energy, ev

Photon energy, ev

FIG. 18. Photoconductivity spectrum of gold-doped germanium for ( A ) p-type germanium with the Fermi level a t the lower acceptor level and ( B ) n-type germanium with the Fermi level at the upper acceptor level [R.Newman, Phys. Rev. 94, 278 (1954)l.

of impurity photoconductivity properties of Au-, Fe-, Ni-, and Co-doped germanium. They find that, for n-type samples, the long wavelength photoionization absorption limit corresponds to the energy separation between the upper acceptor level and the conduction band. For p-type samples, the long wavelength photoconductivity limit corresponds to the energy separation between the lower acceptor level and the valence band. Over the wavelength range where they can obtain such data, impurity photoconductivity in n-type material is shown to involve the generation of free electrons, and in p-type material the generation of free holes. They find, however, that the spectral response curves for n- and p-type samples are appreciably different in character (Fig. 18). Newman suggests that, since the p-type curve exhibits the usual rise to a maximum at the ionization limit, it is due to a single type ionization process; namely, the ionization of holes from M o centers. He suggests further that the form of the n-type curve which does not show the usual rise to a maximum at the ionization limit may be due to the superposition

PHYSICS OF SEMICONDUCTOR MATERIALS

41

of two photoconductive spectra, one corresponding t o the ionization of electrons from M- centers and the other corresponding to the ionization of electrons from M - centers. A more natural explanation for the difference in the forms of the n- and p-type curves may be found in the differences in the Coulomb interaction between the center and the corresponding carrier (80). Thus, in p-type samples the hole inceptor is a neutral center M o

Mo

+ hv+

M-

+ h+

(18)

and there is a Coulomb attraction between the free carrier and the ionized center; whereas in n-type samples, the electron inceptor is a negatively charged center M M= hv + Me(19)

+

+

and there is a Coulomb repulsion between the free carrier and the ionized center. Accordingly, M o will have a larger photoionization cross section a t the ionization limit than M-, due to differences in the wave function of the free carriers. The cross sections will be more alike a t higher photon energies since the wave function of the more energetic free carriers then generated will not be as sensitive to the charge on the ionized center as are the wave functions of lower energy free carriers. An increasing photoionization cross section with increasing photon energies is, accordingly, not unreasonable for M= centers. Morton and co-workers (75) have studied the photoconductivity due to the gold level 0.05 ev above the valence band in p-type germanium, but their measurements did not extend beyond 15 microns so they were unable to reach the photoionization limit. If, as shown by Dunlap (76), this level is a donor, one would expect that the photoresponse near the ionization limit should show a wavelength dependence intermediate between those for the other two levels. The distribution of known impurity levels in the forbidden bands of silicon and germanium are shown schematically in Figs. 19 and 20. The data for silicon are less complete than for germanium due t o the greater difficulties of preparing crystals of high chemical purity. It is also found that the electrical properties of silicon depend markedly on the thermal and growth history (81).Information about the energy levels of impurity centers in silicon have therefore been limited for the most part t o impurities with solubilities greater than lo1*per cm8. At sufficiently low temperatures, germanium containing low concentrations of group III-B or V-B elements exhibits a reversible, nondestructive electrical “breakdown” effect at relatively low (-5 volts/cm) applied electric fields (82).The “breakdown” which manifests itself as a

42

ELIAS BURSTEIN AND PAUL H. EQLI

1.21

Au

In 0.16

FIG.19. Distribution of impurity levels within forbidden band of silicon.

0.785

FIQ.20. Distribution of impurity levels within forbidden band of germanium.

sharp increase in current density of several orders of magnitude is attributed to charge carrier multiplication involving impact ionization of impurities by free carriers which gain the necessary kinetic energy from the applied electrical field. The relatively low fields required for breakdown are attributed to the high mobility of the free carriers in germanium at low temperature and the low ionization energies of the bound carriers. Measurements of Hall effect as a function of applied electric field

PHYSICS OF SEMICONDUCTOR MATERIALS

43

have verified that the breakdown effect is characterized by charge carrier multiplication (83).The effect has also been studied as a function of temperature, background radiation, concentration of various impurities, and magnetic field. The results of these studies showed, among other things, that the discontinuities in the current-voltage curves a t liquid hydrogen temperature reported by Rider (84) are due to charge carrier multiplication rather than to an increase in mobility. In addition, it was found that those parameters which tend to reduce the mobility of the carriers, such as magnetic field and high impurity concentrations, increase the critical breakdown field. The various experimental results are found to be consistent with a simple theoretical treatment of the “breakdown l1 effect (83).According to this treatment the carriers gain energy at a rate, A , under the influence of the applied electric field, and lose energy at a rate, B, by interaction with the lattice. At equilibrium these rates are equal, but the “effective” temperature of the electrons has been increased by the applied field. Breakdown is assumed t o occur when the average energy of the electrons is equal to the ionization energy of the impurities. This leads to a critical field for breakdown which may be expressed in terms of the measured mobility, p , as

where c, is the velocity of sound. This formula yields breakdown fields of the correct order of magnitude and describes qualitatively the way in which the critical field has been found to depend on the mobility and ionization energy. There is relatively little data on impurity levels in gray tin. Busch and co-workers (86) have reported a value of 0.005 ev for donors.* This value is however based on data taken above 60” K where relatively few carriers are bound to donor centers, and is therefore somewhat uncertain. There are a variety of energy levels within the forbidden band of diamond which are relatively deep, in keeping with the relatively low dielectric constant of diamond (e, = 5.7) (86). However, there is no reliable information about the identity of these levels. A similar situation exists for silicon carbide (87). In the MIII-NV compounds, Welker (46) has shown that group 11-B elements, such as Zn and Cd, behave as acceptors and presumably enter into the lattice substitutionally a t MI11 lattice sites; and that group VI-B

* On the basis of a hydrogen model, with to = 47 derived from the energy gap using Moss’ relation (as), i.e., c2E0 = 174, and m* = m, the ioniration energy would be 0.006 ev.

44

ELIAS BURSTEIN AND PAUL H . EQLI

elements such as Se and Te behave as donors and presumably enter into the lattice substitutionally at N v lattice sites. I n addition, group IV-B elements may behave either as donors or as acceptors, apparently depending on whether~theysubstitute a t MI11 or a t N v sites, respectively. We may expect that the ionization energies of the group 11-B, group IV-B, and group VI-B elements in the MlII-Nv compounds can be interpreted qualitatively in terms of a hydrogen model for the un-ionized impurity center. Thus in GaSb, Leifer and Dunlap (88) find that the acceptor impurities in p-type samples containing 10'7 carriers/cm* at room temperature have ionization energies of 0.024 ev. On the basis of a hydrogen model, they calculate an effective mass for holes of 0.35 m which is in agreement with the effective mass value of 0.39m obtained from the geometric mean of the density of states masses. Similarly, in AlSb, Blunt et al. (89) report an energy level at 0.14 ev for Te-doped samples which is in agreement with the value of 0.125 ev calculated for a hydrogen model and an effective mass for electrons equal to m.* No data has as yet been reported on distribution coefficients of the group 11-B, IV-B, and VI-B impurity elements in the MIII-Nv compounds. We may expect, by analogy to silicon and germanium (69), that those elements with tetrahedral covalent radii comparable or smaller than the host element for which they substitute in the lattice, will have relatively large distribution coefficients, while those which have larger radii will be less soluble. The group IV-B elements do not behave as impurity centers when introduced substitutionally as foreign elements in the diamond group of semiconductors. By analogy we may expect that, in the absence of disordering, group 111-B and group V-B elements will also be electrically inactive when introduced substitutionally as foreign elements in the M"WV compound semiconductors. Thus, As and Ga should be electrically inactive when substituted for Sb and In, respectively, in InSb. Information about the electrical behavior of simple lattice defects in silicon and germanium, i.e., lattice vacancies and interstitials, has been obtained primarily from studies of the effect of neutron and electron bombardment on electrical and optical properties (90). James and LarkHorovitz (91) suggest, on the basis of these studies, that the formation of lattice vacancies by the displacement of atoms from their normal site to interstitial positions involves the breaking of covalent bonds and that, consequently, lattice vacancies behave as acceptor centers with relatively

* Hrostowsky et al. (24) observed impurity band conduction in p-type InSb at temperatures below 10" K and obtained an ionization energy for acceptors of 0.007 ev, which corresponds, on the assumption of a hydrogen model, to an effective mass for holes of 0.15m.

PHYSICS OF SEMICONDUCTOR MATERIALS

45

deep levels. They also suggest that interstitial host atoms behave as donor centers and that both the vacancies and the interstitials have two states of ionization. Mayburg (92), who has investigated the effect of heat treatment on germanium under conditions where the effect of chemical impurities such as copper are largely eliminated, suggests that the defects produced by heat treatment are most likely Frenkel defects, i.e., equal number of vacancies and interstitials. Recent studies of the effects of plastic deformation on the electrical properties of germanium (93) indicate that dislocations behave as deep lying acceptors with an energy level 0.2 ev below the conduction band. The compound semiconductors will, by their very nature, exhibit a greater variety of simple defects than the elemental semiconductors. Thus, vacancies can occur a t both the MI1’ and the NV sites, and both the MI1’ and the NV atoms may occupy interstitial positions. Since both types of vacancies involve unpaired electrons, both may be expected to behave as acceptor centers with relatively deep lying levels. I n addition, in the absence of any appreciable polar character, both types of interstitials may be expected to behave as donor centers. The effect of electron and neutron bombardment in the electrical properties of InSb (94) has been interpreted in terms of such lattice defects using the models for lattice defect centers in germanium proposed by James and Lark-Horovitz. Blunt et aE. (89) have observed deep lying levels, 0.75 ev, above the valence band. These they attribute to Sb vacancies in AlSb which are presumably formed as a result of the larger evaporation of Sb from the melt and from the crystal. They also found a level 0.3 ev below the conduction band which they suggest may be due t o interstitial A1 or to other interstitial impurities. The concentration of lattice vacancies is likely t o compounds since, as in the case of the homopolar be small in the MIX1-NV semiconductors with diamond structure, the energy required for the formation of lattice defects may be expected t o be relatively large. An upper limit of 1014per cm3to the equilibrium concentration of lattice defects in InSb is indicated by the fact that a t various laboratories crystals of InSb have been prepared having free carrier concentrations of the order of 1014 per cm8. Although, vacancies and interstitials may not play an important role in determining the normal electrical conductivity of the stoichiometric MI1’-NVcompounds, they may play an important role as “trapping” and “recombination” centers. 3. Polar Semiconductors

Ionic crystals have much higher concentrations of lattice defects than do homopolar crystals and they exhibit much larger deviations from stoichiometry. The available data for the alkali and silver halides (31)

46

ELL48 BURBTEIN AND PAUL H. EGLI

indicate concentrations of 10’8 per cma Schottky defects in NaCl and KCl at the melting point and about 1021per ,ma Frenkel defects in AgC1. I n the alkali halides it is possible t o obtain, by vapor treatment a t elevated temperature, deviations from stoichiometry corresponding t o an excess of 10’9 alkali atoms/cma or 10l8 halogen atoms/cma. Similar magnitudes of deviations from stoichiometry have been obtained in PbS (95), PbSe (96),CdS (9?‘),CuI (98),and other polar semiconductors. Very little is known about the type of defects which are present a t the and MIV-NV1 compounds, but we may infer melting point in the MKK-NV* from the investigations of Wagner (99) that crystals with close packed structures such as the rock salt structure will tend t o favor Schottky defects, whereas crystals having the more open wurtzite and zinc blende structures will tend to favor Frenkel defects. By analogy with the alkali and silver halides (31), we would further expect that a t lower temperatures Schottky defects would condense to form vacancy clusters and dislocations and that Frenkel defects would recombine. The electrical behavior induced by stoichiometry deviations in ionic semiconductors differs from that for homopolar semiconductors. I n ionic crystals, the excess component is incorporated into the crystal as ions having their normal valence. For crystals in which Schottky defects predominate, the ions are incorporated at normal lattice sites and an equivalent concentration of vacancies is created at the lattice sites of the deficient component. The net effect of a deviation from stoichiometry in this case is to introduce electrons and negative ion vacancies when the electropositive component is in excess, or holes and positive ion vacancies when the electronegative component is in excess. * For crystals in which Frenkel defects predominate, the excess component is incorporated as interstitial ions. The extent t o which the composition may be deviated from stoichiometry is determined by the energy required t o incorporate the Corresponding vacancy or interstitial ion and will therefore, in general, be different for excess electropositive and excess electronegative components. It is frequently rather difficult t o obtain stoichiometric crystals of compound semiconductors by normal crystal growth techniques even when these compounds melt congruent1y.t This is due in part t o diffi-

* In ionic crystals such as NiO where the excess electron or hole is highly localized at the ions, there has been a tendency to talk about ions with different valence rather than about free electrons and holes. The expression “controlled valency ” has its basis in this point of view. t A large group of compound semiconductors, particularly the oxides and sulfides of the group 11-Band group IV-B metals, have actually been given the name of “nonstoichiometric ” semiconductors since they were generally available in the form of nonstoichiometric crystals whose properties were determined by deviation from stoichiometry rather than by impurities.

PHYSICS OF SEMICONDUCTOR MATERIALS

47

culties in adjusting the stoichiometry of the melt from which the crystal is grown and in part to the large deviation from stoichiometry which may be tolerated by the crystal. Even if the semiconductor melts congruently, there are two other considerations which bear on the stoichiometry. The first consideration is the incongruent vaporization of the melt or crystal whereby the composition of the vapor is different from that of the melt or crystal with which it is in equilibrium. This incongruent vaporization may occur directly from the solid when the molecule is unstable in the vapor state. It occurs also when the type of binding in the melt is appreciably different from that in the solid, so that effectively the compound is dissociated in the melt. This tendency for incongruent vaporization is small for highly ionic materials where coulombic forces persist in the melt and even in the vapor. NaC1, for example, has a dissociation constant of only 4 X at its boiling point. Dissociation is more complete as the amount of covalent binding increases. Thus a melt of a binary intermetallic compound is essentially a miscible mixture of the two metallic liquids which have appreciably different partial pressures. The component with the lower boiling point will have the higher partial vapor pressure. The composition stability of a melt as measured by the tendency of the melt to lose one constituent preferentially can be roughly predicted by comparing the boiling points of the two elements. For example, in the MII1-NVsystems, the melts of the compounds in which M is indium (b.p. = 2090°C) or gallium (b.p. = 2000°C) are most stable when N is antimony (b.p. = 1440"C), less stable when N is arsenic (b.p. = 604"C), and least stable when N is phosphorus (b.p. = 280°C). Among polar semiconductors the same relation between stability and boiling points can be observed in the PbS group. PbTe, which has the smallest boiling point difference (-35O"C), is the most stable melt and is the easiest from which to grow a crystal. PbS, with the largest boiling point difference (-1450"C), is the least stable melt and the most difficult from which to grow a crystal. The stability of solids with respect t o vapor follows the same trends as the melt-vapor system, and thus the boiling point relations provide a clue as t o the deviations from stoichiometry induced by heating in a vacuum. The solid-vapor relations are even more important in compounds that sublime. The rough indications of dissociation constants that could be inferred from boiling point relations are not adequate t o permit prediction of stoichiometric relations. I n practice, the problem is minimized by maintaining small volume of vapor. The second consideration is the difference in energies required t o form cation and anion vacancies (or interstitials) which causes the invariant point of the solid-liquid system to be displaced from the stoichiometric ratio. Thus, Goldberg and Mitchel (96) have shown that the invariant

48

ELIAS BURSTEIN AND PAUL H. EGLI

point for PbSe occurs for a composition in which there is a n excess of 0.005% Se, a composition which results in p-type crystals. Deviations from stoichiometry which are incurred during crystal growth can generally be modified by subsequent vapor treatment (95, 97). This in fact represents the most practical means for controlling stoichiometry and thereby controlling the electrical behavior of the ionic semiconductors. The concentration of lattice defects in ionic crystals will depend on the impurity content as well as on the deviation from stoichiometry (100). Thus the incorporation of a substitutional or interstitial impurity ion with an ionic charge different from that of the host lattice will be accompanied by the formation of an equivalent number of “charge-compensating ” lattice vacancies or interstitials (101). For example, in crystals where Schottky defects predominate, the incorporation of cations with a larger ionic charge or anions with a smaller ionic charge than the corresponding ions of the host lattice (e.g., Ga+3 or C1- in CdS) will be accompanied by positive ion vacancies. The incorporation of cations with a smaller ionic charge or anions with a larger ionic charge than the corresponding ions of the host lattice (e.g., Ag+ and AS-^ in CdS) will be accompanied by negative ion vacancies. The incorporation of impurities into the lattice may be complicated by aggregation into pairs or larger clusters which may also include lattice vacancies (102). Charge compensation may be brought about without introducing lattice defects by the simultaneous incorporation of a second impurity (103). In CdS, for example, the incorporation of monovalent cations such as Ag+ may be compensated by the incorporation of monovalent anions (e.g., Cl-) or trivalent cations (e.g., Ga+3). Charge compensation may also be brought about by a simultaneous deviation from stoichiometry (104). Thus, in CdS, the incorporation of monovalent cations such as Ag+ may be compensated by the incorporation of excess sulfur; and the incorporation of monovalent anions such as C1- may be compensated by the incorporation of excess cadmium. Charge compensation by a deviation from stoichiometry differs from the other charge compensation processes in that it has the effect of introducing an equivalent number of free electrons or free holes. I n general, the solubility of impurities with an ionic charge different from that of the ion of the host lattice which it replaces will be determined primarily by the energy required t o create the corresponding charge-compensating lattice defect. The solubility can be increased by incorporating charge-compensating ” impurities or by a charge-compensating ” deviation: from- stoichiometry (106). The converse of the latter also occurs. A greater deviation from stoichiometry can be ((

PHYSICS OF SEMICONDUCTOR MATERIALS

49

obtained by incorporating impurities having the appropriate ionic charge. Thus in PbS, Bi+3 increases the solubility of excess lead and decreases the solubility of excess sulfur; whereas Ag+ produces the opposite effect. The electrical behavior of “simple” impurity centers in the ionic semiconductors is similar in some respects to that of impurities in homopolar semiconductors. Substitutional impurities having a n ionic charge different from that of the host lattice act as traps for either electrons or holes, depending on whether they have an effective positive or a n effective negative charge. Thus cations with a greater positive charge and anions with a smaller negative charge than the ions of the host lattice which they replace (e.g., Ga+3 and C1-l in CdS) have an effective positive charge and act as donor type centers; anions with a greater negative charge and cations with a smaller positive charge than the ions of the host lattice which they replace (e.g., Ag+ and As-3 in CdS) will have an effective negative charge and act as acceptor type centers. Interstitial ions will, in general, also exhibit fairly simple behavior, i.e., the cations will behave as donor type centers and the anions will behave as acceptor type centers. We may also expect th at a hydrogen model will be applicable t o such centers in semiconductors having relatively large dielectric constants (i.e., low energy gaps) providing the potential well a t the impurity atoms is not too large. Under these conditions the impurity centers may be expected to have relatively low ionization energies, given by El = (EH/teff2) (m*/m), where eelf is a n effective dielectric constant which will have a value intermediate between that of the optical and static dielectric constants.* The optical ionization energy will be appreciably larger than thermal ionization energy because of Franck-Condon effects. For ionic semiconductors with low “effective” dielectric constants and large effective masses the impurity centers will exhibit deep lying energy levels. The carriers will accordingly be tightly bound in relatively small orbits, which can be described in terms of a modified hydrogen model having a deep potential well a t t,he impurity atom. The behavior of lattice vacancies as traps for electrons and holes has been fairly well established in simple ionic crystals by the extensive investigation of color centers in the alkali halides (31). Negative ion vacancies have a n effective positive charge and act as traps for electrons t o form F-type centers; positive ion vacancies have a n effective negative charge and act as traps for holes t o form V1-type centers. The properties

* Elements which can exist in more than one stable state of ionization in which the outer (valence) electrons are highly localized, such as the transition metal ions (e.g., manganese which can exist as Mn+*, Mn+* and Mn+9 will tend to behave as deep traps even in high dielectric constant semiconductors.

50

ELIAS BURSTEIN AND PAUL H . EGLI

of such centers have been interpreted qualitatively in terms of a simple model in which the vacancy is treated as a cavity, imbedded in a dielectric medium (106). The cavity is characterized by a potential well whose depth is determined by the Madelung potential, the electron affinity, the polarization of the surrounding medium, and an effective diameter which is essentially determined by the position of the nearest neighbor ions. For crystals with small dielectric constants such as the alkali halides, the wave function of the bound carrier will be determined largely by the potential well and will accordingly be largely concentrated near the vacancy. A fairly good approximation can be obtained for the F- or V1-center in the alkali halides by superimposing the atomic wave functions of the six ions forming the nearest neighbors to the missing ion (lo?').With increasing dielectric constant, the contribution from the potential well may be expected to become less important, the wave function of the bound charge carrier will spread out, and the ionization energy will decrease. For polar semiconductors with high dielectric constants, i.e., small energy gaps, we may expect therefore that a hydrogen model will be applicable. I n crystals made up of divalent anions and divalent cations, the vacancies may exhibit two energy levels corresponding to two states of ionization. Other type electron-trapping and hole-trapping centers may exist in other forms, such as clusters of positive and negative ion vacancies and impurity vacancy pairs. The ionization energies of trapped electrons and trapped holes a t lattice defect centers are found to decrease in going from the alkali halides to the silver halides to the PbS group of semiconductors, in part because of an increase in dielectric constant and in part because of a decrease in effective mass. The available data for the PbS group of semiconductors indicate that the lattice defect centers introduced by stoichiometric deviations, as well as by unidentified impurities which may be present, have relatively shallow energy levels in n- and p-type materials (55, 95), as would be expected from the high values of the dielectric constants. There is unfortunately very little other information about impurity and lattice defect centers in the PbS group of semiconductors. Information about impurity and lattice defect centers is now available for CdS, ZnS, ZnO, and CdTe from semiconductor, photoconductor, and luminescence studies. Kroger et al. (64) have obtained, from conductivity and Hall measurements, ionization energies of 0.02 ev relative to the conduction band for lattice defect centers due to excess Cd in CdS (presumably S- vacancies) and for Ga+3 and C1- centers in CdS, all of which behave as donors. For centers due to excess sulfur, presumably Cd++ vacancies, a value of approximately 1.1 ev relative t o the valence

PHYSICS OF SEMICONDUCTOR MATERIALS

51

band is proposed. Crystals containing Ag+ and Cu+ which behave as acceptors in CdS exhibit luminescence and photoconductive bands beyond the intrinsic luminescence and photoconductive limits. On the assumption that these bands involve the direct ionization and subsequent radiative recapture of electrons, the energy levels due to Ag+ and Cu+ have generally been assigned to positions 0.4 and 0.7 ev above the valence band (97, 108). Bube ( l o g ) , on similar evidence, also arrives at a value of approximately 0.5 ev relative to the valence band for Ag+ and Cu+ in ZnS. More recently, however, Lambe and Klick (110) have suggested on the basis of infrared photoconductivity and quenching studies a t low temperature that the Ag+ centers in CdS have ionization energies of approximately 0.4 ev relative to the conduction band rather than to the valence band. They propose that the luminescence band beyond the intrinsic edge involves the radiative capture of holes rather than electrons and that the photoconductive band just beyond the intrinsic edge involves a two-step process consisting of (a) the optical generation of holes and (b) the subsequent thermal generation of electrons. The latter mechanism is in agreement with the observation that the magnitude of the photoconductive response just beyond the edge decreases rapidly with decreasing temperature (111). Jenny and Bube (46) have recently determined the ionization energies of impurity and lattice defect centers in CdTe. They report an ionization energy of 0.003 ev for I- which behaves as a donor. For acceptor type impurities such as Li+, Ag+, Sb-8, and P-* they obtain ionization energies varying from 0.3 t o 0.5 ev (relative to the valence band). Miller (112)and Harrison (113)have reported ionization energies of the order of 0.02 ev for donor centers due presumably to excess Zn in ZnO. Vines and Maurer have reported an ionization energy of 0.29 ev for acceptor centers due to excess I in CuI. We note thatin the CdS group the ionization energies of donor centers are appreciably smaller than those of the acceptor centers. This is due to differences in the character of the neighboring ions. Thus, the atomic wave function at the cations which are the nearest neighbors for donor centers are generally less localized than the atomic wave function at the anions which are the nearest neighbors for acceptor centers. The ionization energies of donor centers in CdS appear to be in accord with a hydrogen model (64).The much higher ionization energy of the acceptor centers indicates either a deep potential well a t the center and/or a large effective mass for holes. The decrease in ionization energies of acceptors with increasing atomic number in going from CdS to CdSe to CdTe may be attributed to a decrease in effective mass of holes and an increase in dielectric constanst.

52

ELIAS BURSTEIN AND PAUL H. EGLI

V. LIFETIMEOF ELECTRON-HOLE PAIRS(114) Various events may occur when electron-hole pairs are created by photoionization or by injection a t point contacts or p-n junctions. The electron-hole pair may recombine directly either by nonradiative recombination processes with the emission of phonons-the inverse of thermal ionization by phonons; or by radiative recombination processes with the emission of photons-the inverse of thermal ionization by photons associated with the black body radiation corresponding t o the temperature of the semiconductor. A third possibility is a three-body recombination process in which an electron and hole recombine and a third free carrier is excited to higher energies. This corresponds either to Auger recombination (116) or to the inverse of impact ionization by free carriers (116). The electron-hole pair may also recombine indirectly via recombination or “deathnium” centers. This involves the initial trapping of the minority charge carriers by centers to which majority charge carriers are already bound, followed by the trapping of majority charge carriers. * The capture of free carriers a t trapping or recombination centers must, of course, also involve some means of removing the excess energy. This may be accomplished by the emission of phonons or photons or by the excitation of another free carrier. The lifetime of electrons and holes need not be equal. Thus, the minority charge carriers may be temporarily captured by trapping centers from which they are thermally re-excited before recombination, by subsequent capture of majority charge Carriers, can take place. The minority charge carriers may actually be trapped and thermally re-excited a number of times before ultimate recombination with the majority charge carriers takes place. Both types of carriers may be temporarily captured by “trapping” centers from which they are thermally re-excited. The difference in behavior of ‘(trapping” centers and “recombination” centers in semiconductors is due to differences in the capture cross section of the centers for minority and majority charge carriers. Thus recombination centers have an appreciable capture cross section for both minority and majority charge carriers, whereas the trapping center has an appreciable capture cross section for the minority carriers and only a very small capture cross section for the majority charge carrier. Small capture cross sections are generally associated with Coulomb repulsion between the trapping center and the free carrier. Semiconductor materials may be characterized by two lifetimes: a

* Many of the concepts concerning “trapping centers” have been developed during the course of luminescence and photoconductivitystudies (117).

PHYSICS OF SEMICONDUCTOR MATERIALS

53

“diffusion” lifetime, and a “conductivity” lifetime. * The diffusion lifetime, which has also been called the “minority charge carrier lifetime” and “free pair lifetime,” is the average period of time during which the electrons and holes are simultaneously free and can therefore participate in diffusion phenomena. It is this lifetime which determines the properties of p-n junctions, transistors, and other devices which depend on the diffusion of “minority” carriers. The conductivity lifetime, on the other hand, is the lifetime during which a t least one type charge carrier remains free to carry current. It includes the time during which one member of the electron-hole pair (generally the minority charge carrier) may spend in a bound state a t a trapping center. It is this lifetime which determines the magnitude of the response of intrinsic photoconductors and other devices which depend on a change in conductivity. The conductivity lifetime may be equal to or larger than the diffusion l3etime.t The method used to determine lifetimes depends on the character of the material, and on the lifetime which is of interest. Conductivity lifetimes may be measured directly by a determination of either the time constant or the magnitude of the intrinsic photoconductive response (118). Diffusion lifetimes may be obtained directly by the familiar diffusion length experiments (119), or may require indirect measurements which involve the determination of the magnitude of the photoelectromagnetic (PEM) effect (120).The PEM effect is essentially a Hall effect produced by diffusion currents. Thus, electron-hole pairs generated a t the surface by photoionization set up a concentration gradient and diffuse into the crystal. Under the influence of a magnetic field perpendicular to the diffusion direction, i.e., Perpendicular to the direction of the incident photoionizing radiation, the two types of carriers are deflected in opposite directions and an emf appears in the third perpendicular direction, The direct measurement of diffusion length is limited to materials in which the “diffusion” lifetime is of the order of lo-’ sec or larger. Accordingly, such measurements have for the most part been practically limited to a relatively small number of semiconductors. The measurement of the response time of photoconductivity can be used, under favorable conditions, for lifetimes of the order of lo-* sec. The determination of the lifetimes from measurements of the magnitude of photoconductive response or from measurements of the PEM effect can be used for lifetimes * From a more fundamental point of view, the lifetimes of interest are the electron

lifetime, the hole lifetime, and the “free pair” lifetime (114). t There are obviously other lifetimes which apply to other phenomena. Thus luminescent materials are characterized by a “luminescence” lifetime. In CdS containing Ag+, Lambe and Klick (110) find that the luminescence lifetime, which presumably corresponds to a hole lifetime, is much shorter than the conductivity lifetime which is effectively the electron lifetime.

54

ELIAS BURSTEIN AND PAUL H. EGLI

of sec and smaller. Both measurements, however, require a knowledge of the flux density of the incident radiation. Some investigators have therefore employed a procedure in which the lifetime of electron-hole pairs is determined from the ratio of the magnitude of the PEM effect to that of the photoconductive response (121). This procedure does not require a knowledge of the flux density of the incident radiation; however, it may lead to large errors unless the diffusion and photoconductivity lifetimes are equal. I n germanium, the different methods for measuring bulk lifetimes of excess electron-hole pairs yield essentially the same value a t room temperature, i.e., the conductivity and the diffusion lifetimes are equal. The lifetime, however, is found to be a strongly structure sensitive property which depends on the thermal and mechanical history and composition of the sample. It appears, therefore, that the lifetime is determined by “indirect )’recombination a t “recombination centers” in the form of lattice imperfections and chemical impurities. The theory for the indirect recombination of electrons and holes at traps has been discussed by Hall (I%,%’), by Shockley and Read (123), and by Fan (124). From the data on the decrease in lifetime with increase in temperature and increase in free carrier concentration, Hall obtained a value of 0.22 ev for the position of the unidentified recombination centers above or below the conduction band. Burton and his co-workers (78) have shown further that nickel and copper act as recombination centers having properties similar to those of the “recombination” centers reported by Hall. They showed that the lifetime observed for ordinary germanium crystals could be readily accounted for by nickel contamination of the order of 1OI2 per cm8 and copper contamination of the order of 1 0 1 3 per cma. The larger cross section of these impurity centers for holes is consistent with the fact that they behave as acceptor centers. Tyler et al. (126)have reported that Fe and Co also act as recombination centers a t room temperature. Dislocations and other large-order lattice imperfections also behave as recombination centers (93).The density of dislocations in good single crystals of germanium is probably sufficiently small that they do not limit lifetime. The centers produced in germanium by thermal quenching (1.26) or by bombardment with high-energy particles (127) also act as recombination centers. Although the “photoconductive” lifetime is equal t o the diffusion lifetime in germanium at room temperature, it is considerably larger than the diffusion lifetime at liquid nitrogen temperatures. This is attributed to the temporary trapping of minority carriers a t trapping centers (128). Fan et al. (129) find evidence for more than one set of trapping states; with different states being effective in different temperature ranges. They

PHYSICS OF SEMICONDUCTOR MATERIALS

55

also find that, although thermal quenching introduces fewer acceptor states than copper impurity atoms, it can be more effective in reducing lifetime. Newman and Tyler (79) have reported the existence of complex trapping and infrared quenching effects in n-type germanium doped with gold, iron, nickel, and cobalt. These effects were not always observed, however, so that it was not clear whether they were due to the impurity levels or to other type centers. More recent studies of n-type germanium doped with manganese (1SO) show very clearly that the trapping and quenching effects are associated with the impurity levels and that they can be correlated directly with the position of the Fermi level. The absence of the trapping and quenching effect in some of the n-type germanium samples doped with the other impurities is attributed to the fact that the Fermi level was not appropriately located. Shulman et al. (127) have recently demonstrated that high-energy electron bombardment of n-type germanium introduces hole trapping centers, presumably associated with the lattice vacancies and interstitials produced by the bombardment. The diffusion lifetime of available silicon specimens has been increasing steadily as techniques for obtaining good crystals have been improving. Values as high as several milliseconds have been reported. Silicon also exhibits very long conductivity lifetimes at room temperature. This is attributed to the trapping of minority charge carriers at “trapping centers.” Haynes and Hornbeck (131), who have carried out a detailed investigation of the trapping of minority carriers in both n- and p-type silicon, report that there are at least two trapping levels for electrons and at least two for holes. They find that the electron traps have large capture cross sections (1O-I2 cm2)and exhibit multiple trapping, whereas the hole traps have very much smaller capture cross sections (10-17 cm2) and do not exhibit any appreciable multiple trapping. They find further that recombination does take place at the trapping centers, particularly a t high majority charge carrier density and that the rate of recombination is proportional to the square of the majority carrier density. Van Roosbroeck and Shockley (1%) have estimated the rate of direct radiative recombination in germanium by application of the principle of detailed balancing to the thermal equilibrium rate of photon absorption which they determined from data on optical constants at room temperature. They obtain a radiative recombination lifetime of 0.75 sec based on the optical constant data of Briggs (19) which is appreciably larger than the observed time constants.* A similar calculation for silicon (Table XIV) (133) based on the optical data of Fan et al. (17) and Dash et al.

* Calculations based on the more recent optical data of

et al. (19) yield a value of 0.3 eec.

Fan el al. (17)and Dash

56

ELIAS BURSTEIN AND PAUL H. EGLI

(19) yields 3.5 sec. The relatively small rates of radiative recombination in germanium and silicon is due in part t o the weak second-order intrinsic absorption involving nonvertical transition in the region of the long wavelength absorption edge. That direct radiative recombination of electrons and holes does take place has been confirmed by Haynes and Briggs (134)) who were actually able t o measure the spectral distribution of the recombination radiation in silicon and germanium. The experimental data are found t o be in good agreement with theory.* TABLE XIV. Lifetime for Radiative Recombination*

Si Ge PbS PbSe PbTe InSb

1.4 X 10’0 2 . 4 x 1013 2 . 9 X 10l6 2 x 10” 6 X loi6 2 . 2 X 1018

2.0 X 3.7 x 1.4 X 3.3 x 1.8 X 2.6 X

log 1013

loan 1022

1022

3.5 0.3

-10 x 10-6 -3 x 10-1.7 X lo-* -0.4 x 10-6

-10-3

-10-3

9

x

10-6 -0.1 x 10-8

* E. Burstein, S. Teitler, and G. Picus, unpublished.

tR

t TR p

= rate of recombination. =

radiative recombination lifetime.

= highest value reported.

Aigrain (136) has recently reported evidence for recombination radiation involving the “indirect ” recombination of electrons and holes a t traps. The data of Haynes and Hornbeck, which show that the rate of recombination at the “trapping centers” in silicon is proportional t o the square of the charge carrier density, suggest that the capture of majority carriers involves a three-body recombination in which the excess energy is given t o a second majority carrier. Information about the lifetime of excess electron-hole pairs is also available for PbS which is one of the few other semiconductor materials in addition to silicon and germanium in which “efficient ” transistor action has been observed. Moss (190) has determined lifetime of excess electronhole IJairs in PbS by diffusion length, photoconductivity, and P E M measurements on a number of specimens of natural crystals having a wide range of resistivities. The lifetimes ranged from 6 X sec t o 9 X sec. All three measurements could not be made in every case, but in the ranges where it was possible t o do so, the values of the lifetime deter-

* Haynes (134) has recently reported a second radiative recombination emission peak in germanium a t 1.5 microns which is associated with the corresponding rise in absorption reported by Fan et al. (17’) and Dash et al. (19).

PHYSICS OF SEMICONDUCTOR MATERIALS

57

mined by the three methods were found t o be in agreement. We may conclude therefore that the conductivity and diffusion lifetimes were identical and that there was no measurable trapping of minority charge carriers. Moss found further that, under optimum conditions, the lifetime was proportional to the square of the resistivity, which suggested that the recombination of free carriers involved three-body recombination processes (Fig. 21). His results are in agreement with a theoretical prediction by Pincherle (115)that the lifetime in PbS may be determined by three-body

I

0.1 0.01 Resistivity, Qcm

0 11

FIG.21. Electron hole pair lifetimes in PbS as a function of resistivity [T. S. Moss, Proc. Phys. SOC.B66, 993 (1953)l.

recombination processes. Pincherle has carried out an approximate calculation of Auger recombination in PbS which indicates a lifetime equal sec-crn-B/nz, which is larger by a factor of 5 than the experito 3 x mental value obtained by Moss. An extrapolation of MOSS'Sdata to the intrinsic concentrations at 300' K (ni = 3 X 10l6 per cm3) indicates a lifetime for intrinsic PbS of approximately 60 psec. The intrinsic radiative recombination lifetime calculated from the optical constant data of PbS by means of the Roosbroeck-Shockley theory is found to be -1 X sec (Table XIV). This value is somewhat smaller than MOSS~S extrapolated value for intrinsic material. The radiative recombination process should therefore presumably determine the lifetime in good crystals of intrinsic PbS. Since the radiative recombi-

58

ELIAS BURSTEIN AND PAUL H. EOLI

nation lifetime is inversely proportional to the charge carrier density while the three-body recombination lifetime is' inversely proportional t o the square of the charge carrier density, three-body recombination should be the dominant recombination mechanism in impure crystals. The luminescence which has been reported a t the long wavelength edge in PbS (136) probably arises from the radiative recombination of electron-hole pairs. The lattice defects in PbS have shallow energy levels and therefore do not behave as trapping centers or as recombination centers. Consequently, in spite of the relatively high density of lattice defects which are present even in good stoichiometric crystals, the observed minority carrier lifetimes are in agreement with the theoretical lifetimes for direct recombination. The value of 300 psec (based on photoconductivity measurements) which has been reported for the lifetime in PbS films undoubtedly represents a conductivity lifetime rather than a diffusion lifetime and is very likely due to the trapping of the minority charge carriers. There is as yet little information about the lifetime of excess electronhole pairs in PbSe and PbTe. The radiative recombination lifetimes calculated from the optical constant data are found to be -2.5 X sec for intrinsic PbSe and -3.4 X 10-6 sec for intrinsic PbTe (Table XIV). Information about the lifetime of excess electron-hole pairs in the group M1ll-B-NV-B compounds is available only for InSb, InP, and InAs. From diffusion length and photoconductivity measurements on InSb, Moss obtains a value for the diffusion lifetime of 2.5 X 10-8 sec at room temperature for a sample which was intrinsic down to 220' K. The response time increased with decrease in temperature so that he was also able to obtain an approximate value of 0.8 X sec for the photoconductive response time a t liquid air temperature. Kurnick et al. (121) have obtained a value of lO-'sec for the lifetime of InSb at liquid nitrogen temperature from data on the ratio of the PEM effect to the photoconductive current. This value is in reasonable agreement with that of Moss at low temperatures. Using the optical constant data of Avery, Goodwin, Lawson, and Moss (137) it is possible to calculate the radiative recombination lifetime for intrinsic InSb a t room temperature by means of the Roosbroeck-Shockley theory. The radiative recombination lifetime is found to be of the'order of 5 X 10-7 sec (Table XIV).* Welker has observed transistor action in InP and has reported a value

* Kurnich and co-workers (private communication) have recently obtained lifetime of -0.1 X 10-0 sec a t room temperature. It appears that the lifetime is due to direct (presumably radiative) recombination. This is plausible since any levels within the narrow forbidden energy gap would be essentially completely ionized at room temperature.

PHYSICS OF SEMICONDUCTOR MATERIALS

59

of 400 microns for the diffusion length. Talley and Enright (138) obtain a value of 7 X 10-6 sec for the lifetime in InAs. There is no detailed information about the factors which determine the lifetime in the M1ll-B-Nv-B compounds. Both recombination and surface recombination lifetimes have been found to be quite ‘‘structure’’ sensitive, as in the case of silicon and germanium. The effect of deviating from stoichiometry on lifetime is not known. Lifetime data are also available for semiconductors such as CdS and CdSe (116, 139). I n these materials the holes are rapidly captured at trapping centers, 80 that the observed lifetime values correspond to conductivity lifetimes. The trapping centers are found to have capture cross sections of 10-l6to 10-18 cm2for holes, as compared to capture cross sections as small as 10-21 cm2for electrons. These materials also exhibit fairly complex photoconductive and luminescent phenomena which are associated with the presence of deep lying trapping and recombination centers. Rose (116) suggests that these and several related phenomena could be understood in terms of the properties of two classes of trapping centers. On this basis he presents a reasonable explanation for superlinear photoconductivity, infrared quenching, and photoconductive sensitization. FOR DETERMINING CHARACTERISTIC PROPERTIES VI. PROCEDURES OF

SEMICONDUCTORS

1. Carrier Concentration and Mobility

For the “simple energy band” type of semiconductors, the free carrier concentration and mobilities may be obtained fairly unambiguously from conductivity and Hall effect measurements. At temperatures where the free carriers arise predominantly from impurity or defect levels within the forbidden band, the conductivity and isothermal Hall coefficients are given simply by u = nep

R lRlo

= +r/en = rp

(21)

where n is the concentration of the majority free carriers, p is their mobility, and r is a statistical factor which depends on the nature of the scattering mechanism. The magnitude of r is equal to 3 ~ / 8for free carriers Fvhich obey classical statistics and have a mean free path independent of their energy as in lattice scattering, whereas r is equal to 1.93 for free carriers having a mean free path proportional to the square of their energies, as in the “Conwell-Weisskopf” type of scattering, and equal to 1 for free carriers obeying Fermi statistics. It should be noted that the expression for the Hall coefficientis based on the condition that ( p H ) << 1.

60

ELIAS BURSTEIN AND PAUL H. EGLI

The Hall mobility of the majority carriers is given by the product JRlo and, except for uncertainty about the scattering mechanism and therefore about the factor T , the free carrier concentration is obtained from the value of R. When diffusion lengths of the minority carrier are appreciable, as in the case of silicon and germanium, data can be obtained for the conductivity mobility of the minority carriers from “drift” type experiments (140). By carrying out conductivity measurements as a function of temperature in the impurity “exhaustion” range of temperatures in which the energy levels within the forbidden band are completely ionized and R remains constant, one can obtain information about the temperature dependence of the “conductivity mobility” of the “majority” charge carriers from conductivity temperature dependence data. Willardson et al. (141) have recently shown that, for semiconductors with degenerate energy bands, the Hall coefficient depends in a more complicated way on the carrier concentration and is appreciably dependent upon the magnetic field. I n p-type germanium where lattice scattering dominates, the Hall coefficient is given by the expression

where npl and np2are the concentrations of holes in the V1 and V z valence bands, respectively, and p p l o and pPzo are the corresponding weak magnetic field mobilities. I n the limit of high fields, the Hall coefficient is given by the expression

Similar expressions also apply for the weak- and high-field magnetoresistance in semiconductors with degenerate energy bands, having spherical energy surfaces. By taking into account the relative concentrations and mobilities of the holes in the V 1and V zvalence bands, Willardson el al. are able to account for the magnitude, temperature dependence, and magnetic field dependence of Hall and magnetoresistive effects in high purity p-type germanium. I n the “transition” range of temperatures in which the free carriers arise from both extrinsic and intrinsic ionization processes and in the “intrinsic” range of temperatures in which the carriers arise primarily from intrinsic ionization processes, the determination of free carrier concentrations from Hall effect and conductivity measurements is no longer straightforward even in the “simple energy band” type semiconductor. In the transition range of temperatures, the conductivity and Hall coef-

61

PHYSICS OF SEMICONDUCTOR MATERIALS

ficients are given by u = nnepn

r

IR/ = -

+ n p e p p = epp(nnb + n,)

nnpAZ- n,pp2

lel (nnpn

lRb =

V

=

r nnb2 - np

+ n p ~ p ) ~I4 ( n n b +

nnb2 - np P nnb + n p

np)’

(24)

where b is the mobility ratio pn/pp. I n the intrinsic range of temperatures where nn = n, = nil these expressions reduce to 6 =

lRla

niepp(b

+ 1)

= rpp(b - 1)

For semiconductors having a very large electron t o hole mobility ratio as in the case of InSb and InAs, the expression for the conductivity and Hall coefficient in the intrinsic regions simplify to u

= niepn

lRIu = rpn Under these conditions, the determination of ni and p, is fairly straightforward. Various procedures may be followed to obtain information about the free carrier concentration and mobilities in the transition range and intrinsic range of temperature (1.42).These procedures are based essentially on different methods for obtaining information about the mobility ratio and depend largely on the nature of the semiconductor and the type of specimens which are available. They include (a) the determination of the electron and hole mobilities as a function of temperature in the impurity range for both n- and p-type material and extrapolating these values t o the temperature of interest, (b) the determination of mobility ratio at the Hall inversion temperature (where R = 0 and nnpn2 = n p p p 2 ) from the ratio of the value of the conductivity extrapolated from the “exhaustion” range of temperatures (uexh)t o the conductivity a t the inversion temperature (ai.,) which is given by uerb/uinv= (1 b)/b, and (c) the determination of the mobility ratio a t the Hall reversal temperature (where nnpn = n p p p )from the ratio of the maximum value in the Hall curve when it reverses sign (Rmsx)t o the value of the Hall coefficient in the exhaustion range ( R o )which is given by R,,,/Ro = ( b - 1)2/4b. I n obtaining values for the concentration of free carriers and their

+

62

ELIAE) BURSTEIN AND PAUL H. EGLI

dependence on temperature in the transition and intrinsic ranges of temperatures, it is usually assumed in the absence of other information that the mobility ratio is independent of temperatures, and also that n, - np is independent of temperature in the transition region. It is now generally recognized that the mobility ratio in most semiconductors does vary with temperature and that values for the free carrier concentration and mobility, and therefore the thermal energy gap, derived on the assumption of a constant mobility ratio, are generally in error. Because of these complications, there are only a small number of semiconductors for which we now have good charge carrier and mobility data in the transition and intrinsic ranges of temperature. These include silicon and germanium, which are available in the form of relatively pure single crystals, and InSb and InAs, which exhibit high mobility ratios. Approximate values of the mobilities of the free carriers may also be obtained from magnetoresistance measurements in the impurity range of temperatures. In simple energy type two-band model semiconductors, the transverse magnetoresistance is independent of orientation and is given by (143) - -- 0.38 X 1O-l6(rH)' (27)

*' U

In practice the mobilities obtained from magnetoresistance measurements do not agree with Hall and conductivity mobilities. This is generally attributed to the fact that the energy bands have more complicated structures than is assumed in the two-band model. As shown by Willardson et al. (1.41))even when the energy bands have spherical energy surfaces, differences between the magnetoresistance mobility and the Hall and conductivity mobilities may arise when the bands are degenerate. Measurements of the optical absorption by free carriers may be used to obtain carrier concentrations.* However, such measurements are restricted to relatively high concentrations so that the method is limited in applicability. 2. Forbidden Energy Gap

Information about the forbidden energy gap and energy levels within the forbidden band is most readily obtained from (a) free carrier concentration versus temperature data derived from Hall effect and conductivity measurements, which yield values for the thermal energy gap, and (b) from optical absorption and photoconductivity versus wavelength data

* Such measurement8 have recently been used by Fan, Shepherd, and Spitzer (17) to determine the free carrier concentration in germanium over the intrinsic range of temperature and have yielded values of K G = n,n, in agreement with those derived from Hall and conductivity measurements.

PHYSICS OF SEMICONDUCTOR MATERIALS

63

which yield values for the optical energy gap. The thermal method is limited in practice to semiconductors with energy gaps less than about 2 to 3 ev because of difficulties introduced by the high temperatures which are needed to get into the intrinsic range. a. Thermal Energy Gap. The values of n, and np calculated from the Hall effect and conductivity measurements are used to obtain empirical expressions for the intrinsic ionization equilibrium constant of the form

K Q = nnnp= AT3 exp ( - E / k T )

(28)

which is of the same form as the theoretical expression for KG given by equation (2). The difference between the coefficient A in the empirical expression and the corresponding coefficient in the theoretical expression for K , based on m, = mp = m, is generally attributed either to a temperature dependence of the thermal energy gap or to effective masses of the free carriers different from m, or to both effects. It is also usually assumed that the thermal energy gap has a linear temperature dependence, i.e., EG = EGO - PT, where /3 = (aEa/dT),, so that the theoretical expression may be written as

K Q = N , N . exp ( - P / k ) exp (- E G O / ~ T )

(29)

The value of E in the empirical expression (28) is accordingly identified with EGO, where EGO is the value of EG extrapolated to T = 0, on the assumption that the linear dependence of EQ,holds down to T = O.* The values.of A and EGO are obtained fairly readily from n,np data by plotting k In (n,np/T9) versus 1/T. Over the range of temperatures under consideration, this generally yields a straight line whose slope is equal to EGO, and whose intercept at ( 1 / T ) = 0 is equal to In A . The curve will, however, deviate from a straight line when EGO has a higher order dependence on temperature. It is necessary to employ Fermi statistics when the Fermi level lies close to the valence or conduction band. This situation occurs, for example, when the energy gap is comparable to kT, or when the energy gap is somewhat larger than kT but the ratio of the effective masses of the free carriers differs considerably from unity as in the case of InSb (146). b. Optical Energy Gap. The intrinsic optical absorption constants are generally quite high and, except for measurements in the region of the long wavelength absorption limit, it is necessary to use very thinsections for absorption measurements or to resort to reflection measurements

* Recent measurements reported by Macfarlane et al. (144) for Ge indicate that, below 20' K, the variation of EQ with lifetime goes to zero. Effectively EQ' is the constant part of EQ a t the high temperature.

64

ELIAS BURSTEIN AND PAUL H. EGLI

(146'). Reflection measurements on uncontaminated surfaces generally yield fairly accurate data in the region of strong absorption, but are somewhat inaccurate in the region of weak absorption, i.e., in the region of the absorption edge. Transmission measurements, on the other hand, yield accurate data in regions of relatively moderate or weak absorption. * By combining transmission measurements with reflection measurements, it is possible to obtain fairly complete absorption spectra. The determination of the optical energy gap, Eo, from data a t the long wavelength intrinsic absorption or photoconductivity edge involves an uncertainty as to how to extrapolate the data to zero absorption in the absence of an adequate theory for the shape of the absorption edge. Different investigators have therefore used a variety of criteria for choosing values for the optical energy gap from the data at the long wavelength edge. Values are reported which are based on a particular choice of the magnitude of the absorption constant (17),or on the position of maximum slope in the absorption versus wavelength curve (147),etc. Macfarlane et at. (144) have recently employed an expression for the shape of the ionization edge in crystals where the absorption process involves nonvertical transitions, and have used this expression to extrapolate the data to zero absorption. The determination of the temperature dependence of the optical energy gap from data on the variation of the absorption edge with temperature may also be somewhat uncertain since there may be an appreciable change in absorption cross section as well as a change in energy gap. This is discussed by Fan, Shepherd, and Spitzer (17)for silicon and germanium. Despite these complications, the determination of the energy gap by optical measurements has in fact proven to be more reliable than Hall effect and conductivity measurements for a number of semiconductors such as the PbS family (cf. Section VII). Optical absorption measurements are particularly useful for obtaining approximate values of the energy gaps of new materials since they can be carried out on relatively impure, polycrystalline samples which would otherwise be unsatisfactory for Hall and conductivity measurements. It is, moreover, the most practical method for obtaining information about energy gaps greater than 2 ev. It is generally desirable where possible to carry out both photoconductivity and absorption measurements in order to confirm the fact that the long wavelength absorption edge is associated with transitions from the full band to the bottom of the conduction band. It is also important to

* Recently, techniques have been developed a t the G. E. Research Laboratory t o prepare single-crystal thin sections of the order of 1 micron in thickness, so tha t Dash et al. (19) were able to measure absorption constants of the order of lo4 cm-1 by transmission techniques.

PHYSICS OF SEMICONDUCTOR MATERIALS

65

verify that the absorption edge does not vary from specimen to specimen (i.e., that it is an intrinsic property of the semiconductor). I n this connection, it is desirable, in order to obtain the best possible value for the optical energy gap, to carry out the optical measurements on high purity materials t o avoid absorption by free carriers which would otherwise mask the intrinsic absorption at the extreme absorption edge. In practice, there are still relatively few materials available in sufficiently pure form to enable data to be obtained below intrinsic absorption constants of 100 cm-I. Without such data, there will be some uncertainty about the determination of the optical energy gap, since there is always the possibility of additional structure in the absorption edge at absorption constant values below 100 cm-', particularly in crystals whose intrinsic absorption edge involves nonvertical transitions, as in the case of germanium. As shown by Braunstein (193) information about, the intrinsic absorption edge and therefore about the optical energy gap can be obtained from radiative recombination emission measurements even when the intrinsic absorption edge is masked by free carriers. Radiative recombination emission studies may therefore yield better estimates of the optical energy gap of impure materials than do transmission measurements. * Thus Braunstein obtains a value of 1.1 ev for the optical energy gap of GaAs from radiative recombination emission data whereas the optical transmission measurements of Barrie et al. (152) have only yielded an upper limit value of 1.35 ev. 3. Energy Levels within the Forbidden Band

a. Thermal Ionization Energies. The theoretical expression for the ionization equilibrium constant in n-type material involving the generation of electrons from donor levels with the forbidden band having only one bound state for electrons takes the form

where ni and nu are the concentrations of ionized and un-ionized donor levels, fi and fu are the partition functions per unit volume of the ionized and un-ionized centers, EDis the ionization energy of the donor level, and where No, the effective density of states in the conduction band is given by 2(2mzkT/h2)'*.The equilibrium constant involving the generation of holes is given by a similar expression. Data for determining E D are best obtained at temperatures where most of the charge carriers are bound to the center. Under the conditions

* Photoconductivity measurements may also be expected to yield better estimates

of the optical energy gap when the absorption edge is masked by free carriers.

66

ELIAS BURSTEIN AND PAUL H. EGLI

nn << nu, nu = N D - N A , and n1 = N A , where N D is the total concentration of ionized and un-ionized donors, and N A is the concentration of compensating acceptor levels, the dependence of carrier concentration on temperature is given by*

The value of E D is determined from a plot of k In (n,/T”/”)versus 1/T. The existence of compensating centers has only fairly recently been taken into account in the analysis of temperature dependence of the carrier concentration. Many of the earlier determinations of the ionization energies are based on the use of an expression for the conductivity of the form u = A exp (- E/2kt), which effectively ignores the existence of compensating centers. When excited states are taken into account the equilibrium between the bound and ionized states of the impurity centers is determined by the equilibrium constant (67).

K D = N , / C ga exp (- E,/kT)

(32)

9

where E , is the energy of the qth state measured relative to the conduction band edge, and gp is the corresponding degeneracy of the state. I n ~ an impurity center in which terms of the equilibrium constant K D for only the lowest energy state El is taken into account, the equilibrium constant can be written

The factor within the brackets, which corrects for the presence of higher states, becomes negligible for temperatures where kT is appreciably less than the energy separation E z - E l between the ground and first excited states. Since in practice these are the conditions under which ionization energies are obtained from carrier concentration versus temperature data, no error is introduced by the use of K I in obtaining ED. b. Optical Ionization Energies. Optical absorption by impurity or lattice defect centers may involve (a) photoionization involving transitions from the ground state to the conduction band or to the valence band and (b) optical excitation involving transitions from the ground state to excited states. The optical absorption due to impurity and lattice defect centers is

* The dependence on temperature is considerably more complicated when other centers having different ionization energies are present.

PHYSICS OF SEMICONDUCTOR MATERIALS

67

usually quite small because of their low concentrations. Essentially the same information is obtained from optical absorption and from photoconductivity measurements. This is due to the fact that the impurity photoconductivity measurements are generally carried out under conditions of small absorption constants, corresponding to low impurity concentrations. Under these conditions the photoconductive response per unit incident photon is proportional to the absorption constant, so that a spectral response curve is equivalent to a plot of the photoionization absorption constant versus wavelength. The photoconductivity measurements can, however, be carried out for impurity concentrations which are too small to be investigated by optical absorption and are therefore particularly valuable for the weakly soluble impurities. In addition, the photoconductivity measurements for a single specimen will yield data covering several decades of the absorption constant, which can be obtained only by the use of several specimens with a range of thicknesses or impurity concentrations in the case of optical absorption measurements. On the other hand, optical measurements require temperatures only low enough for the majority of the carriers to be bound to impurity centers, whereas conductivity measurements require considerably lower temperatures. Thus, optical absorption data can be obtained a t liquid nitrogen temperature for an impurity with an ionization energy of 0.05 ev, but require liquid hydrogen temperatures, or lower, for photoconductivity measurements. As in the case of the intrinsic absorption edge, the determination of optical ionization energies from data at the photoionization absorption or photoconductive edge is somewhat arbitrary, since there is as yet no adequate theory for the shape of the impurity photoionization absorption spectra. A better estimate of the ionization energy can frequently be obtained when excitation bands exist.

4. Effective Mass For semiconductors which can be described by a simple energy band structure, there is a single effective mass which enters into the various properties. This is no longer true for semiconductors with complicated energy band structures. For such materials the effective mass is a tensor quantity, being defined in terms of the energy-momentum curvatures as (148)

Different properties involve the "curvature

"

mass in different ways

(1.49). Thus it is necessary to use a different quantity for the effective

mass in the expressions for mobility, density of states, high frequency

68

ELIAS BURSTEIN AND PAUL H. EGLI

dielectric constant, etc. The two most often encountered are the “density of states mass” and the (‘inertial mass.” The density of states mass is defined by the equation for the density of states

N(E)

=

47r

(2m(N))’m[ElS

(35)

where N ( E ) is the number of states with energies in the range from E to E dE and E is measured relative to the band edge. For the two-band model, m ( N )= m*. When the surface of constant E consists of a single ellipsoid, m“) equals the geometric mean of the three principle curvature masses mii; i.e., nCN) = (m,imjimkk)M.When there are M equivalent ellipsoids, N ( E ) d E is M times that for one ellipsoid, and m(*) is equal to MS‘j (mi;mj3mhk)$4. The inertial mass m(’), which is the quantity involved in the real and imaginary parts of the high-frequency dielectric constant, is defined as the average over all the carriers of the ratio of the force applied to the rate of change of velocity. * For single or multiple ellipsoidal energy surof faces, (m(I))-lis given by the arithmetic mean of the curvatures the plot of energy against momentum in the principle directions, i.e. (mU))-l = m..-lm..--Im a% 33 kk-‘

+

For energy bands having spherical energy surfaces which are degenerate, (m(I))-lis given by an arithmetic mean weighted by the fraction of the carriers in the degenerate bands. For the valence bands in silicon and germanium, (mcr))-l is given by

where npl/n, is the fraction of carriers in the V1valence band and npl/np is the fraction of the carriers in the V Zvalence band; i.e., the hofes in the two bands behave as independent carriers. Information about the curvature masses can be obtained directly from cyclotron resonance experiments (12’).The application of cyclotron resonance is, however, limited to semiconductors in which wcr > 1, where 1 / r is the collision frequency for the free carriers and w c is the angular rotation “cyclotron resonance” frequency given by wc = (eH)/m*c. Cyclotron resonance experiments have thus far been carried out for silicon and germanium and more recently for InSb (160). For homopolar or weakly polar semiconductors where EG = Eo, it is possible to obtain a value for the geometric mean of the density of states

* The mobility involves a combination of the density of states mass and the inertial mass. For nonpolar lattice scattering, the mobility is proportional to (m(’))-1(m(N))-%.

PHYSICS OF SEMICONDUCTOR MATERIALS

69

masses from the room temperature value for the optical energy gap and the room temperature value of K = nnnp by means of equation (28). When greater reliance can be placed on the temperature coefficient of the optical energy gap, the value for the geometric mean of the density of states mass can be calculated from the empirical value of A and Po = (aEo/aT),by means of equation (29). The geometric mean effective masses obtained in this way for silicon and germanium are found t o be in good agreement with the values calculated for the effective masses determined by cyclotron resonance measurements (17). For semiconductors with simple energy bands, the individual values of m, and mp can be obtained from the ratio of the mobilities and the geometric mean mass by making use of the theoretical relation between the lattice scattering mobilities and effective masses (60)

(mn/mp)-6”d (37) Density of states masses can also be obtained from a combination of thermoelectric power and Hall effect measurements which yield information about the dependence of the Fermi level on free carrier concentration. This procedure has been applied t o germanium (151), GaAs (166),PbSe and PbTe (153), and more recently to InSb. Information about the inertial mass may be obtained from microwave measurements of the real and imaginary parts of the dielectric constant, or from microwave measurements of the magnetoconductivity. Such measurements have been carried out for germanium (154) and indium antimonide (155). Mn/pp

=

VII. CURRENTINFORMATION ON FAMILIAR SEMICONDUCTOR MATERIALS Although considerable progress has been made in understanding the physics of semiconductors and in the experimental techniques for measuring the properties of semiconductors, precise data on the fundamental semiconductor properties are available for only a relatively small number of semiconductor materials. This is due primarily to the fact that there are as yet only a small number of semiconductor materials which can be obtained in the form of well-defined single crystals of relatively high purity and perfection. 1. Elemental Semiconductors

a. Diamond. Approximate information about the forbidden energy gap is available from the long wavelength absorption edge which occurs at 0.22 microns, corresponding t o an energy gap of 5.6 ev (156). Since the absorption processes at the edge may involve exciton formation rather

70

ELIAS BURSTEIN AND PAUL H. E a L I

than photoionization, this value represents a lower limit t o the forbidden energy gap. The impurity and defect centers in diamond have relatively deep energy levels so that essentially all of the carriers are bound a t room temperature (157). In order to obtain mobility data, it has been necessary to carry out measurements on carriers generated by photoionization from impurity or defect centers using ultraviolet radiation (i68).The relative densities of free electrons and holes produced by photoionization depend on the distribution of donor and acceptor impurities within the particular specimen under investigation. Redfield (I59) has nevertheless been able to obtain reasonable estimates of the individual lattice scattering mobilities of electrons and holes by carrying out measurements on many specimens. Three of the diamonds had a negative Hall effect which obeyed a T-W temperature dependence. Since these diamonds had the highest mobilities (and differed from one another by relatively small values), Redfield concluded that the values of the electron mobility in these diamonds (pn = 1800 cm2/volt-sec) corresponds approximately t o the lattice scattering mobility. The data for holes were somewhat more uncertain. One of the samples with a positive Hall effect was found t o have a mobility which approached a T-” temperature dependence at high temperatures. Redfield therefore concluded that the mobility for this sample ( p p = 1200 cm2/volt-sec) is not too far from the true lattice mobility for holes. b. Silicon and Germanium.* The latest and probably most reliable value for the lattice scattering mobilities of electrons and holes and the thermal energy gap of silicon is that of Morin and Maita (180). By combining measurements of the conductivity in the exhaustion range of temperatures with values for the drift mobility at 300’ K measured by Prince (i61),they obtain expressions for the temperature dependence of the lattice scattering mobilities of electrons and holes of the formt

The temperature dependence of the lattice scattering mobilities depart from the T-l.s law expected for nonpolar scattering by acoustical modes in a simple energy band model. This is attributed in part to scattering by optical modes and in part to effects associated with the structure of the energy bands. By combining extrapolated lattice scattering

* For a more detailed discussion see article by H. Brooks,

and Electron Phys. 7, 85 (1955).

Advances in Electronics

t The expression for lattice scattering conductivity of electrons involves corrections for impurity scattering which was present in the n-type samples. Impurity scattering was found to be negligible in the p-type samples above 150’ K.

PHYSICS OF SEMICONDUCTOR MATERIALS

71

mobilities with intrinsic conductivity data, they obtain an empirical expression for K G of the form

KG = nnn,

= 1.5

x

1033T-3exp (-1.21/kT)

(39)

indicating a value of EGO = 1.21 ev. The latest reported value for the lattice scattering mobilities and the thermal energy gap of germanium is also due to Morin and Maita (162). They obtain empirical expressions for the lattice scattering conductivity mobilities for electrons and holes of the form

They suggest that the departure of the electron mobility from a T-" dependence could conceivably be due to scattering by optical modes, but that in the case of the hole mobility it must be attributed to effects associated with the structure of the energy bands. They obtain an empirical expression for K of the form

K G = n,n,

= 3.10 X 1032T-Sexp (-0.785/kT)

(41)

indicating a value of EGO = 0.785 ev.* c. Gray Tin. Progress in the elucidation of the semiconductor properties of gray tin have paralleled improvements in the techniques for preparing suitable specimens for investigation (163) and for stabilizing gray tin at temperatures above the transition point (164). The early data of Busch et al. (165) indicated a value of approximately 0.1 ev for the thermal energy gap. Busch and co-workers (85, 166) have since reported a value of 0.08 ev based on conductivity measurements and a similar value based on magnetic susceptibility measurements. Kendal (16?'),on the other hand, has reported a value of 0.064 ev based on conductivity measurements. Ewald (168) and Becker (163) have reported values of 0.088 ev and 0.085 ev, respectively, from Hall and conductivity measurements. Becker has also been able to observe intrinsic photoconductivity in gray tin at low temperatures and has obtained thereby an approximate value of 0.075 ev for the optical energy gap. Busch and Wieland (85) obtain a T-n temperature dependence for the mobility of electrons p, =

6.3 X 1O6T-sncm2/volt-sec

(42)

* The earlier values of E~Oforsilicon and germanium were based on the assumption of a T-S temperature dependence of the lattice scattering mobilities.

72

ELIAS BURSTEIN AND PAUL H. EGLI

They also obtain a mobility ratio which decreases with increase in temperature, indicating a higher temperature dependence for the mobility of holes. Becker's values for the electron mobility are appreciably higher than those reported by Busch and co-workers. Thus he obtains a value of 2 X loa cm2/volt-sec a t 273°K as compared to Busch and Wieland's value of 1.4 X lo3 cm2/volt-sec.* d. Tellurium. Although much of the data for tellurium is based on polycrystalline specimens, the more recent investigations have been carried out on single crystals. Tellurium has a hexagonal structure and therefore exhibits anisotropic conductivities. It is somewhat unusual in that it exhibits two Hall reversals. Thus relatively pure tellurium is a p-type semiconductor at low temperatures. It reverses to n-type at about -233" K, depending on the impurity concentration, and then reverses back at a temperature of about 500" K, independent of impurity concentration. The thermal energy gap is found to be 0.32 ev (170).At 300" K the intrinsic resistivity is 0.29 ohm-cm in a direction parallel to the c-axis and 0.56 ohm-cm in a direction perpendicular to the c-axis. Tellurium exhibits two long wavelength optical absorption edges, one a t 0.32 ev for radiation polarized perpendicular to the c-axis, and one at 0.37 ev for radiation polarized along the c-axis (171). The shift in the absorption edge with temperature for radiation polarized parallel to the c-axis is found to be -2 to - 5 X ev/deg, and approximately the same value is found for radiation polarized perpendicular to the c-axis. t Bardeen's analysis of Bridgeman's data (173) on the effect of pressure on the conductivity of tellurium shows that the energy gap increases with increase in lattice constant and indicates a value of 1.7 X ev/deg for the lattice dilation contribution to (aEG/aT).$ It appears therefore that the direct temperature contribution to the temperature dependence of the energy gap is negative and only slightly larger than the lattice dilation contribution. Several mechanisms have been proposed to account for the two reversals. These include (a) a temperature dependent mobility of electrons due to the existence of two overlapping conduction bands having different curvatures a t the band edge (l76'), (b) the generation of lattice defects, presumably tellurium vacancies, which behave as accep-

* More recent measurements by Becker (189) indicate deviation from a T-B temperature dependence for mobility of electrons. t Moss (172) has reported a positive temperature coefficient based on photoconductivity measurements on thin films. Such measurements, however, are generally complicated by other factors. $ Neuringer (174) has recently carried out a direct determination of the effect of pressure on the absorption edge radiation polarized parallel to the c-axis and has obtained a value of approximately 2 X lo-' ev/dcg, in agreement with the vaIue obtained from the effect of pressure on conductivity.

PHYSICS OF SEMICONDUCTOR MATERIALS

73

tors (177),* and (c) changes in the relative widths of the valence and conduction bands and therefore of the effective masses and mobilities of electrons and holes. The latter mechanism, which is based on a simple energy band model for tellurium developed by Callen (178), also accounts for the anisotropic optical properties of tellurium and for decrease of the lower and upper Hall reversal temperature with increase in pressure (179). 2. Compound Semiconductors

a. PbS Group (180). Putley (56) has carried out Hall effect and conductivity measurements on single-crystal specimens of PbS, PbSe, and PbTe. The electron and hole mobilities in the three materials are found to have a T-2.6temperature dependence in the extrinsic range of temperatures. The magnitudes of the mobilities were, however, found to have a considerable scatter so that it was not possible t o determine the mobility ratio from the data. Values of the mobility ratio were therefore obtained from the maximum negative value of the Hall coefficients. This yielded mobility ratio values of 2.6 for PbS and PbTe and 1.6 for PbSe. Assuming the mobility ratio to be independent of temperature, Putley obtained values of 1.17 ev, 0.50 ev, and 0.63 ev for the thermal energy gaps of PbS, PbSe, and PbTe, respectively, from the slope of the In R versus 1/T curves for temperatures in the range of 500" K to 800" K. These thermal energy gaps are found to be considerably higher than the room temperature optical energy gaps of 0.30 ev, 0.22 ev, and 0.27 ev which were determined from Gibson's optical absorption data extrapolated t o zero absorption (180). The position of the optical absorption edge corresponds reasonably well with the position of the photoconductivity limit for single crystals and films. This, together with the high value of the absorption coefficient a t the absorption edge and its insensitivity to impurity content, serves as evidence that the optical energy gaps constitute intrinsic values. Scanlon (181) has more recently been able to obtain intrinsic conductivity in PbS a t temperatures below 500" K by using specimens having small charge carrier concentrations. He obtains a value of EGO = 0.34 ev based on charge carrier concentration versus temperature data which were determined from Hall effect and conductivity measurements on the assumption of a constant mobility ratio of b = 1.4. Although much lower than Putley's value for PbS, his value is still appreciably higher than the optical value obtained by Gibson. It is now generally accepted that the values of the energy gaps for PbS, PbSe, and PbTe obtained from optical measurements are essentially

* Middleton (176) has found evidence for high temperature slope corresponding to an activation energy of E = 0.76 ev.

74

ELIAS BURSTEIN AND PAUL H. EGLI

correct. Smith (180) suggests that values obtained from Hall measurements are in error because (a) the mobility ratio is not independent of temperature, i.e., it appears to increase with temperature, (b) (8Eo/8T) is not constant over the temperature range concerned, and (c) there appears to be a third generation of "extrinsic " carriers. The PbS group of semiconductors differs from the other well-known semiconductors in having a positive temperature coefficient of the energy gap. This manifests itself as a shift of the absorption edge to longer wavelengths as the temperature is decreased. Gibson's optical data yield a value of 8Eo/dT = f4 X lo-* ev/deg for all three semiconductors at room temperature. Bell et al. (.21),have carried out calculations of the dependence of band edges on lattice constant and obtain a value of aE/aT = +2 X ev/deg for the contribution to the temperature dependence from the change in lattice constant. It would appear therefore that the lattice change is the dominant factor in the temperature dependence since broadening due to thermal vibrations yields a negative contribution. 6 . MrI1-NV Compounds. Considerable progress has been made in the study of the semiconductor properties of the M"'-NV compounds in the relatively short time since their semiconductor properties were first reported. Fairly reliable semiconductor data are now available for a number of the MI1'-Nv compounds which can be prepared in the form of single crystals, and particularly for InSb and InAs whose lattice scattering mobility ratios are high. Because of the high electron t o hole mobility ratio in InSb, it is possible to obtain an empirical expression for K = nnnpfairly unambiguously from conductivity and Hall effect measurements. From optical transmission measurements on InSb which was intrinsic at room temperature, Tanenbaum and Briggs (18.2) obtained a value of 0.18 ev from the room temperature optical energy gap and a temperature coefficient of -3 X ev/deg. By combining this value with data for K = nnnpobtained from conductivity and Hall effect measurements, Tanenbaum and Maita (183) obtained a value of (mnmp)s= 0.083 m for the geometric mean density of states mass.* This, when combined with the ratio of the lattice scattering mobilities pn/pp = 80, and the assumption of a T-?6temperature dependence, yields values of 0.2m and 0.03m for the effective masses of holes and electrons, respectively. Madelung and Weiss (146) have recently carried out fairly precise Hall and conductivity measurements for InSb. Using Fermi statistics

* Tanenbaum and Maita were essentially the first to use optical energy gap data together with data for K = nnn, to obtain values for the geometric mean density of states mass.

PHYSICS OF SEMICONDUCTOR MATERIALS

75

they obtain a value of EGO = 0.27, in agreement with the value calculated from data on the optical energy gap a t 300" K and its temperature dependence. They obtain a T-l.6 temperature dependence for the lattice scattering mobility of electrons and a somewhat higher temperature dependence for holes. Hrostowski and co-workers (24) similarly obtain a T-1.7 temperature dependence for the lattice scattering mobilities of electrons. They report impurity band conduction in p-type InSb below 10" K and obtain an ionization energy for acceptors of 0.007 ev. From magnetoconductive measurements in the microwave region, Dexter and Lax (166) obtain values of 0.02m and 0.3m for the inertial effective masses of electrons and holes. On the other hand, Dresselhaus and co-workers (160) obtain a lower value of 0.013m for the effective mass of electrons from cyclotron resonance. More recently, Frederikse and Mielczarek (194) have obtained a value of 0.014m for the effective mass of electrons from thermoelectric power measurements. Tanenbaum and Briggs (188) have reported that the optical absorption edge in InSb shifted to shorter wavelengths with increasing impurity content (183). This anomalous dependence of the optical absorption edge on impurity content occurs primarily in n-type material, but has also been found in strongly p-type material (184). As shown by Burstein (186)and Moss (147),this is due to the very small effective mass of electrons and is associated with a small effective density of states and with the small degeneracy concentration ( N o = 1.6 X 1017 per cm* and Nden = 1.2 X 10'' per cm8 at 300" K). Thus InSb becomes degenerate a t relatively low electron densities. The optical absorption therefore involves vertical transition from the filled band to the lowest unfilled levels in the conduction band. EO should be equal t o EG for nondegenerate samples and approximately equal to E o (1 m,/m,)(E, - E, - 4kT) in degenerate n-type samples. For degenerate p-type InSb, on the other hand, EO should be equal t o EG (1 m,/m,)(E. - Ef - 4 k T ) . On the basis of this model, the experimental data on the dependence of the absorption edge on charge carrier concentration yields a value of 0.03m for the effective mass of electrons, which is somewhat larger than the value of 0.013m obtained from cyclotron resonance. The discrepancy between the two values may be due to an increase in effective mass with increasing energy and to the fact that the optical value is obtained from data for degenerate samples in which the electron states under consideration are well above the bottom of the conduction band, whereas the cyclotron resonance value is obtained at low temperatures for nondegenerate samples in which the electrons under consideration are close to the bottom of the bapd. Fairly reliable data have now been obtained for InAs. The mobility

+ + + +

76

ELIAS BURSTEIN AND PAUL H. EGLI

ratio is found to be of the order of 150 so that it is possible to determine fairly unambiguously from Hall and conductivity measurements the dependence of K = nnnpon temperature. From such data Folberth et al. (186) obtain a value of 0.47 ev for EGO. By combining this with the value of 0.35 ev for the optical energy gap at 300" K, they obtain a value of ev/deg for aE/aT which agrees with the value of - 4 X -4.5 X ev/deg obtained by Talley et al. (158) from transmission. They also obtain a value of O.lm for the geometric mean density of states mass. Their mobility data show indications of a T-W temperature dependence for electrons, but an appreciably higher temperature dependence for holes. The high mobility ratio and the small geometric mean effective mass indicates a relatively small effective mass for electrons. Applying the twoband model relation between the mobility ratios and the effective mass ratios (equation 37) yields as a rough approximation m, = 0.07m and m, = O.lm. As a consequence of the small effective mass of the electrons, InAs exhibits a shift of the absorption edge to shorter wavelengths with increasing electron concentration (138, 184). The magnitude of the shift is, however, much smaller than that for InSb. For AlSb, various investigators have reported a value of approximately 1.6 ev based on conductivity measurements and the assumption that the mobility ratio is independent of temperature. Blunt et al. have also obtained a room temperature optical energy gap of 1.52 ev with aEo/aT = -3.5 X ev/deg. I n general the specimens which were available for investigation were relatively impure, so that it was not possible to obtain information about the lattice scattering mobilities of the carriers or the mobility ratio. Conductivity and Hall measurements for GaSb have yielded values of 0.7 t o 0.8 ev for EGO on the assumption that the mobility ratio is independent of temperature. Leifer and Dunlap (88) obtain a value of 0.80 ev. By combining this value with Newman's value of 0.71 ev for the optical energy gap a t room temperature, they obtain aEo/aT = - 3 X lO-*ev/deg and a geometric mean density of states mass of 0.28m. From this value of mass and the value of 5 for the ratio of electron to hole mobility determined from data at the Hall reversal temperature, Leifer and Dunlap calculate the individual effective masses using the simple two-band model relation between the mobility ratio and effective mass ratios. They obtain m, = 0.2m and mp = 0.38m. Their mobility data for holes indicate a T-'6 scattering law at high temperatures. Blunt et al. (187) have recently reported similar results. They obtain EGO = 0.77 ev and b = 5.5 from Hall and conductivity measurements. From optical measurements they obtain a room temperature optical energy gap of 0.67 ev and a temperature coefficient of -3.5 X ev/deg. For GaAs, Barrie and co-workers (166)obtain a value of 1.35 ev for

PHYSICS O F SEMICONDUCTOR MATERIALS

77

the optical energy gap from transmission data and a temperature coefficient of -5 X 10-4 ev/”C. They also obtain a value of 1.25 ev for the optical gap from photoconductive response data. In addition, they estimate a value of 0.03m for the effective mass of electrons from thermoelectric power measurements on n-type samples containing from 2.5 X 1017 to 1.3 )( 10’8 carriers/cm3. Braunstein (193) obtains 1.1 ev for the optical energy gap from radiative recombination emission data. There have been few measurements of the semiconductor properties of the other MII1-NVcompound semiconductors. Values of the energy gaps have, however, been estimated from optical transmission data for I n P (188), GaP (189), and Alp and AlAs (190). c. M2I1-NIVCompounds. The measurements of Robertson and Uhlig (33) indicate an energy gap of 0.2 ev for MgzSn. More recently Busch and Winkler (191) have made a more intensive study of this family of compounds. From Hall and conductivity measurements and the assumption of a temperature independent mobility ratio they obtain values of 0.77 ev, 0.74 ev, and 0.36 ev for thermal energy gaps EGO of MgzSi, MgzGe, and MgzSn, respectively. They have also obtained energy gap values for solid solutions of MgzSn and MgzGe. Their data indicate a T-4$ temperature dependence for the electron and hole mobilities. From the temperature dependence of the thermoelectric power, they obtain values of ev/deg and dEa/aT = -3.8 X dEa/dT = -8.9 X ev/deg for MgzGe and MgzSn, respectively.* Blunt and Frederikse (192) have recently carried out a detailed study of the optical and electrical properties of MgzSn. They obtain a value of 0.33 ev for E G O . They obtain essentially the same value for EO a t liquid helium temperature from transmission measurements and a somewhat smaller value from photoconductivity measurements. They also obtain a value of -3.5 X 10-4 ev/deg for dEo/dT which is in agreement with the value for dEG/dT obtained by Busch and Winkler. Their data also indicate a T-2;2 temperature dependence for the mobility of both electrons and holes. ACKNOWLEDGMENT Many people in the Crystal Branch have made substantial contributions to the material presented here. The authors are particularly grateful to Bertha Henvis for her perceptive editing and improvements to the coherence of the entire manuscript, to Gerald Picus for stimulating discussions and constructive suggestions and to William Zimmerman for assembling data and editing chemical portions of the review. REFERENCES 1. See earIier reviews of semiconductor materials by Busch, G., 2. angew. Math. u. Phys. 1,3,81(1950);Lark-Horovitz, K., “The New Electronics” in “The Present State of Physics,” p. 57. AAAS, Washington, D.C., 1954.

* These results are at variance with those obtained for MgZSn b y MacFarlane and co-workers from optical data.

78

ELIAS BURSTEIN A N D PAUL H. EQLI

2. Seits, F., “The Modern Theory of Solids,” Chapters 1 and 13. McGraw-Hill,

New York, 1940. 3. Stillwell, C. W., “Crystal Chemistry.” McGraw-Hill, New York, 1938; Wells, A. F., “Structural Inorganic Chemistry.” Oxford Univ. Press, New York, 1945; Euken, A., “Lehrbuch der Chemischen Physik.” Akademische Verlagsgesellschaft, Leipsig, 1949; Winkler, H. G. F., “Structur und Eigenschaften der Kristalle.” Springer, Berlin, 1950; Hume-Rothery, W., “The Structure of Metals and Alloys.” Institute of Metals, London, 1936. 4. Mott, N. F., and Gurney, R. W., “Electronic Processes in Ionic Solids,” Chapter 3, p. 65. Oxford Univ. Press, New York, 1940. 5. De Boer, J. H., and Verwey, E. J. W., PTOC. Phys. SOC.(London)49,extra part 59 (1937);Verwey, E. J. W., in “Semi-conducting Materials” (H. K. Henisch, ed.), p. 151.Academic Press, New York, 1951;Mott, N. F., “Progress in Metal Physics,” Vol. 3, p. 76. Interscience, New York, 1952. 6. Shockley, W., “Electrons and Holes in Semiconductors,” p. 174.Van Nostrand, New York, 1950. 7 Fowler, R. H., “Statistical Mechanics,” 2nd ed., Chapter 2. Cambridge Univ. Press, London, 1936. 8. James, H. M., “Note on the Statistical Mechanics of Semiconducting Crystals with Electron-Vibration Coupling,” presented a t Atlantic City Conference on Photoconductivity (1954);Brooks, H., Advances in Electronics and Electron Phys. 7, 85 (1965). 9. Seitz, F., “The Modern Theory of Solids,” Chapter 8, p. 326. McGraw-Hill, New York, 1940. 10. Mott, N. F., and Gurney, R. W., “Electronic Theories in Ionic Solids,” Chapter 5,p. 160. Oxford Univ. Press, New York, 1940;Callen, H. B., Phys. Rev. 79, 533 (1950). 11. Shockley, W., Phys. Rev. 78, 173 (1950);79, 191 (1950);90, 491 (1953);Pearson, G. L., and Suhl, H., ibid. 85,768 (1951);Abeles, B., and Meihoom, S., ibid. 96,31 (1954);Smith, C.S.,ibid. 94, 42 (1954);Briggs, H. B., and Fletcher, R. C., ibid. 91, 1342 (1953). 12. Dresselhaus, G., Kip, A. F., and Kittel, C., Phys. Rev. 92, 827 (1953);Lax, B., Zeiger, H. J., Dexter, R. N., and Rosenblum, E. S., ibid. 93,1418 (1954);Dexter, R. N., Zeiger, H. J., and Lax, B., ibid. 96,557 (1954);Lax, B.,Zeiger, H. J., and Dexter, R. N., Physica 20, 818 (1954);Dexter, R. N., Lax, B., Kip, A. F., and Dresselhaus, G., Phys. Rev. 96, 222 (1954); Dresselhaus, G.,Kip, A. F., and Kittel, C., ibid. 98, 368 (1955). 1s. Herman, F., Phys. Reu. 88, 1210 (1952); 93, 1214 (1954); Herman, F., Phvsica 20, 80 (1954);Herman, F., and Callaway, J., Phys. Rev. 89, 518 (1953). 14. Dresselhaus, G., Kip, A. F., and Kittel, C., Phys. Rev. 96, 568 (1954);Elliot, R. J., ibid. 96, 266, 280 (1954);Kahn, A. H., ibid. 97, 1647 (1955). 16. Herman, F., Phys. Rev. 96, 847 (1954). 16. Johnson, F. R., and Christian, S. M., Phys. Rev. 96, 560 (1954);Levitas, A., Wang, C. C., and Alexander, €3. H., ibid. 98, 846 (1954). 17. Fan, H. Y . , and Becker, M., in “Semi-Conducting Materials” (H. K. Henisch, ed.), p. 132. Academic Press, New York, 1951;Fan, H. Y., Shepherd, M. L., and Spitzer, W., “Infrared Absorption and Energy Band Structure of Germanium and Silicon,” presented a t the Atlantic City Conference on Photoconductivity (1954);see also Macfarlane, G. G., and Roberts, V., Phys. Rev. 97, 1714 (1955). 18. Hall, L. H., Bardeen, J., and Blatt, F. J., Phys. Rev. 96,559 (1954);Bardeen, J., Blatt, F. J., and Hall, L. H., “Indirect Transitions from the Valence to the Conduction Bands,” presented a t Atlantic City Conference on Photoconductivity

PHYSICS O F SEMICONDUCTOR MATERIALS

79

(1954); see also Macfarlane, G. G., and Roberts, V., Phys. Rev. 97, 1714 (1955). 19. Pfestorf, G., Ann. Physik 61, 906 (1926) (Si); Moss, T. S., Proc. Phys. SOC. (London) B66, 141 (1953) (Si); Brattain, W. H., and Briggs, H. B., Phys. Rev.76, 1705 (1949) (Ge); Briggs, H. B., J . Opt. SOC.Amer. 42, 686 (1952) (Ge); Avery, D. G., and Clegg, P. L., Proc. Phys. Soc. (London) B66,512 (1953) (Ge); Gebbie, H. A., Thesis, Reading University, 1952 (Ge); Dash, W. C., Newman, R., and Taft, E. P., Bull. Am. Phys. SOC.SO, No. 1 (V7), 53 (1955) (Si and Ge). 20. Herman, F., Phys. Rev. 88, 1210 (1952). 21. Bell, D. G., Hum, D. M., Pincherle, L., Sciama, D. W., and Woodward, P. M., Proc. Roy. SOC.A217, 71 (1953). 22. Gibson, A. F., Proc. Phys. SOC.(London) B66, 378 (1952); Avery, D. G., ibid. B64, 1087 (1951); B66, 133 (1953); B67, 2 (1954). 63. Tanenbaum, M., and Pearson, G. L., Phys. Rev. 91,244 (1953); Tanenbaum, M., Pearson, G. L., and Feldmann, W. L., ibid. 93, 912 (1954). 24. Hatton, J., and Rollin, B. V., Proc. Phys. SOC.(London) A67, 385 (1954); Hrostowski, H. J., Morin, F. J., Geballe, T. H., and Wheatley, G. H., Bull. Am. Phys. SOC.SO, No. 3 (GlO), 26 (1955). 66. Blount, E., Callaway, J., Cohen, M., Dumke, W. P., and Philips, J., Phys. Rev. In press. 86. Slater, J. C., and Koster, G. F., Phys. Rev. 94, 1498 (1954). 87. Kittel, C., Bull. Am. Phys. SOC.SO, No. 2 (J3), 19 (1955); Herman, F., Bull. Am. Phys. Soc. SO, No. 3 (DA4), 20 (1955). 28. Slater, J. C., and Shockley, W., Phys. Rev. 60, 705 (1936). 29. Ewing, D. H., and Seitz, F., Phys. Rev. 60, 760 (1936). 30. Steel, E. L., Thesis, Cornell University, 1952. 51. Seitz, F., Revs. Mod. Phys. 23, 328 (1951); 26, 7 (1954). 32. Krumhansl, J. A., “Photoexcitation in Ionic Crystals,” presented at Atlantic City Conference on Photoconductivity (1954). 33. Robertson, W. D., and Uhlig, H. D., A . I . M . E . 180, 345 (1949); J. Electrochem. Soc. 96, 27 (1949). 34. Welker, H., Scientia Electrica 1, part 4, 1 (1954); Physica 20, 893 (1954). 36. Keyes, R. W., Phys. Rev. 84, 367 (1951). 36. Busch, G., and Vogt, O., Helv. Phys. Acta 27, 241 (1954). 37. Goodman, C. H. L., Proc. Phys. SOC.(London) B67, 258 (1954). 38. Moss, T. S., Proc. Phys. SOC.(London) A64, 590 (1951). 39. Briggs, H. B., Cummings, R. F., Hrostowski, H. J., and Tanenbaum, M., Bull. Am. Phys. SOC.28, No. 6 (A9) 8 (1953). 40. Szigetti, B., Trans. Faraday SOC.46, 155 (1949). 4 f . Dunlap, W. C., and Wattera, R. L., Phgs. Rev. 92, 1396 (1953). 42. Srigetti, B., Proc. Roy. SOC.A204, 51 (1950). 43. Burstein, E., and Oberly, J. J., Phys. Rev. 78, 642 (1950); Collins, R. J., and Fan, H. Y., ibid. 93, 674 (1964); Lax, M., and Burstein, E., ibid. 97,39 (1955).

44. Picus, G. S., and Burstein, E., unpublished results.

46. Welker, H., Z . Naturjorsch. 7a, 744 (1952); 8a, 248 (1953). 46. Jenny, D. A., and Bube, R. H., Phys. Rev. 96, 1190 (1954). 47. Wlerick, G., private communication. 48. Seraphin, B., 2.Naturjorsch. 9a, 450 (1954). @. Frohlich, H., Advances i n Phys. 4, 325 (1954). 60. Low, F., and Pines, D., Phys. Rev. 91, 193 (1953).

80

ELIAS BURSTEIN AND PAUL H . EGLI

61. Meijer, H. J. G., and Polder, D., Physicu 19, 255 (1953). 62, Redfield, A. G., Phys. Rev. 94, 537 (1954). 65. Haynes, J. R., and Shockley, W., Phys. Rev. 82, 935 (1951); Brown,

F. C., ibid. 97, 355 (1955). 54. Kroger, F. A., Vink, H. J., and Volger, J., Physicu 20, 1095 (1954). 65. Putley, E. H., Proc. Phys. SOC.(London) B66, 993 (1952); B68, 22 (1955). 66. Petritz, R., and Scanlon, W. W., Phys. Rev. 97, 1620 (1955). 67. Morin, F. J., Phys. Rev. 93, 1195, 1199 (1954). 58. Dunlap, W. C., Jr., Phys. Rev. 96, 40 (1954) (zinc); 97, 614 (1955) (gold). 69. See also review paper, “Impurity Centers in Germanium and Silicon,” by Burton, J. A., Physicu 20, 845 (1954). 60. Conwell, E. M., Proe. I.R.E. 40, 1327 (1952). 61. Pearson, G. L., and Bardeen, J., Phys. Rev. 76, 865 (1949). 6% Bethe, H., M.I.T. Radiation Laboratory Report No. 43-12 (1942). 69. Slater, J. C., Phys. Rev. 76, 1592 (1949). 64. Geballe, T. H., and Morin, F. J., Phys. Rev. 96, 1085 (1954). 65. Morin, F. J., Maita, J. P., Shulman, R. F., and Hannay, N. B., Phys. Rev. 96, 833 (1954). 66. Burstein, E., Oberly, J. J., Davisson, J. W., and Henvis, B. W., Phys. Rev. 82, 764 (1951); Fan, H. Y., and Becker, M., ibid. 78, 178 (1950); Burstein, E., Oberly, J. J., and Davisson, J. W., ibid. 89,331 (1953); Rollin, B. V., and Simmons, E. L., Proc. Phys. SOC.(London) B66, 995 (1952); B66, 161 (1953); Burstein, E., Davisson, J. W., Bell, E. E., Turner, W. J., and Lipson, H. G., Phys. Rev. 93, 65 (1954). 67. Burstein, E., Bell, E. E., Davisson, J. W., and Lax, M., J. Phys. Chem. 67, 849 (1953). 68. Burstein, E., Picus, G. S., Henvis, B. W., and Lax, M., Bull. Am. Phys. SOC.30, No. 2 (FLO), 13 (1955); Burstein, E., Picus, G., and Sclar, W., “Optical and Photoconductive Properties of Silicon and Germanium,” presented a t Atlantic City Conference on Photoconductivity (1954). 69. Burstein, E., Picus, G. S., and Henvis, B. W., unpublished. 70. Koster, G. F., and Slater, J., Phys. Rev. 96, 1167 (1954). 71. Kittel, C., and Mitchell, A. H., Phys. Rev. 96, 1488 (1954); Lampert, M., ibid. 97, 352 (1955); Luttinger, J. M., and Kohn, W., ibid. 96, 802 (1954). 7.9. Kohn, W., and Luttinger, J. M., Phys. Rev. 97, 1721 (1955); Kleiner, W. H., ibid. 97, 1722 (1955). 75. Kohn, W., Phys. Rev. In press. 74. Tyler, W. W., private communication. 76. Morton, G., Hahn, E., and Schultz, L., “Impurity Photoconductivity in Germanium,” presented a t the Atlantic City Conference on Photoconductivity (1954). 76. Dunlap, W. C., Jr., Bull. Am. Phys. SOC.30, No. 2 (F2), 12 (1955). 77. Burton, J. A., Hull, G. W., Morin, F. J., and Severiens, J. C., J . Phys. Chem. 67, 853 (1953). 78. Morgan, T. M., Ph. D. Thesis, University of Illinois, 1954. 79. Newman, R., Phys. Rev. 94,278 (1954); Newman, R., and Tyler, W. W., ibid. 96, 882 (1954); Tyler, W. W., Newman, R., and Woodbury, H. H., ibid. 97, 669 (1955); Tyler, W. W., and Newman, R., Bull. Am. Phys. SOC.SO, No. 1 (V4), 53 (1955); Tyler, W. W., Newman, R., and Woodbury, H. H., Phys. Rev. 98, 461 (1955); 8ee also Kaiser, W., and Fan, H. Y., ibid. 93, 278 (1954).

PHYSICS OF SEMICONDUCTOR MATERIALS

81

80. Burstein, E., and Picus, G., unpublished. 81. Fuller, C. S., Ditaenberger, J. A., Hannay, N. B., and Buehler, E., Phys. Rev. 96,

833 (1954).

8d. Sclar, N., Burstein, E., Turner, R. J., and Davisson, J. W., Phys. Rev.

91, 215 (1954); Sclar, N., Burstein, E., and Davisson, J. W., ibid. 92,858 (1953); Ryder, E. J., Ross, I. M., and Kleinman, D. A., ibid. 96,1342 (1954); DarneU, F. J., and Friedberg, S. A., Bull. Am. Phys. SOC.30, No. 1 (28), 39 (1955); Koenig, S. H., and Gunther-Mohr, G. R., Bull. Am. Phys. Sac. 29, NO. 7 (G3), 13 (1954). 83. Sclar, N., and Burstein, E., unpublished. 84. Ryder, E. J., Phys. Rev. 90,766 (1953); Conwell, E. M., ibid. 90,769 (1953); 94, 1068 (1954). 85. Busch, G., and Wieland, J., Helv. Phys. Actu 26, 697 (1953). 86. Bull, C., and Garlick, G. F. J., Proc. Phys. SOC.(London) A63, 1283 (1950). 87. Busch, G., Helv. Phys. Acta 19, 167 (1946); Busch, G., and Labhart, H., ibid. 19,463 (1946). 88. Leifer, H. N., and Dunlap, W. C., Jr.,Phys. Rev. 96,51 (1954). 89. Blunt, R. F., Frederikse, H. P. R., Becker, J. H., and Hosler, W. R., Phys. Rev. 96,578 (1954). 90. Lark-Horovitz, K., in “Semi-Conducting Materials” (H. K. Henisch, ed.), p. 47. Academic Press, New York, 1951; Cleland, J. W., Crawford, J. H., Jr., and Pigg, J. C., Phys. Rev. 99, 1170 (1955); Klontz, E. E., and Lark-Horovitz, K., ibid. 86, 643 (1952); Brown, W. L., Fletcher, R. C., and Wright, K. A., ibid. 92, 591 (1953); Fan, H. Y., Kaiser, W., Klontz, E. E., Lark-Horovitz, K., and Pepper, R. R., ibid. 96, 1087 (1954). 91. James, H. M., and Lark-Horovitz, K., 2. physik. Chem. 198, 107 (1951). 9.9. Mayburg, S., Phys. Rev. 96, 38 (1954); Sylvuniu Technologist 7, 109 (1954). 93. Pearson, G. L., Read, W. T., Jr., and Morin, F. J., Phys. Rev. 93,666 (1954); Read, W. T., Phil. Mag. (71 46, 775, 1119 (1954); Lipson, H. G., Burstein, E., and Smith, P. L.,Phys. Rev. 99, No. 2, 444 (1955). 94. Cleland, J. W., and Crawford, J. H., Jr., Phys. Rev. 93, 894 (1954); 96, 1177 (1954). 96. Bebrick, R. F., and Scanlon, W. W., Phys. Rev. 96,598 (1954). 96. Goldberg, A. F., and Mitchel, G. R., J . Chem. Phys. 22, 220 (1954). 97. Kroger, F. A., Vink, H. J., and Boomgaard, J. van den, 2.physik. Chem. 203, l(1954). 98. Vine, H., and Maurer, R.,2.Physik 198, 147 (1951). 99. Wagner, C., 2.physik. Chem. B22, 181 (1933); see also Huntington, H. B., and Seitz, F.,Phys. Rev. 61,315 (1942). 100. See review paper by Kroger, F. A., and Vink, H. J., Physicu 20, 950 (1954). 101. Koch, E., and Wagner, C., 2.physik. Chem. B38, 295 (1937). 102. Burstein, E., Oberly, J. J., Henvis, B. W., and Davisson, J. W., Phys. Rev. 81, 459 (1951). 103. Kroger, F. A., and Hellingman, J. F., J . Electrochem. Soe. 93, 156 (1948); 96,68 (1949). 104. Verwey, E. J. W., Haaijman, P. W., Romeijn, F. C., and van Oosterhout, G. W., Philips Research Repts. 6, 173 (1950); Selwood, P. W., Moore, T. E., Ellis, M., and Wethington, K., J . Am. Chem. SOC.71, 693 (1949). 105. Wagner, C., J . Chem. Phys. 18, 69 (1950); Wagner, C., J . Phys. Chem. 67, 3 (1953); Wagner, C., J . Electrochem. SOC.99,346 (1952). 106. Krumhansl, J. A., and Schwartz, N., Phys. Rev. 89, 1154 (1953).

82

ELIAS BURSTEIN AND PAUL H. EGLI

107. Inui, T., and Uemura, Y., Progr. Theorlt. Phys. (Japan) 8, 252 (1950); Kahn, A. H., and Kittel, C., Phys. Rev. 89, 315 (1953). 108. Klasens, H. A., J . Electrochem. SOC.100, 72 (1953). f 0 B . Bube, R. H., Phys. Rev. 90, 70 (1953). 120. Lambe, J. L., and Klick, C. C., Phys. Rev. In press. 111. Lambe, J. L., private communication. 112. Miller, P. H., Jr., Phys. Rev. 60, 890 (1941). f l 3 . Harrison, S. E., Phys. Rev. 98, 52 (1954). 114. See also discuseion of lifetime by Rose, A., Phys. Rev. 97, 322 (1955). f 16. Pincherle, L., unpublished work, discussed in reference 180. f f 6. Frohlich, H., and O’Dwyer, J., Proc. Phys. SOC.(London) A68, 81 (1950); Sclar, N., and Burstein, E., Phys. Rev. 08, 1757 (1955). I f ? . See, for example, Rose, A., RCA Rev. 12, 362 (1951). 118. Goucher, F. S., Phys. Rev. 78, 816 (1950). 119. Valdes, L. B., Proc. I.R.E. 40, 1420 (1952). 110. Moss, T. S., Proc. Phys. SOC.(London) B66, 993 (1933); Moss, T. S., Physica 20, 989 (1954); Moss, T. S., Pincherle, L., and Woodward, A. M., Proc. Phys. Soc. (London) B66, 743 (1953). 261. Kurnick, 8. W., Strauss, A. J., and Zitter, R. N., Phys. Rev. 94, 1791 (1954). 162. Hall, R. N., Phys. Rev. 83, 228 (1951); 87, 387 (1952). 118. Shockley, W., and Read, W. T., Jr., Phys. Rev. 87, 835 (1953). f.24. Fan, H. Y., Phys. Rev. 92, 1424 (1953). 116. Tyler, W. W., Woodbury, H. H., and Newman, R., Phys. Rev. 94, 1419 (1954). 196. Logan, R., Phys. Rev. 01, 757 (1953). 127. Rappaport, P., Phys. Rev. 94, 1409 (1954); Shulman, R. G., Fletcher, R. C., and Brown, W . L., Phys. Rev. 96, 833 (1954). 118. Navon, D., Bray, R., and Fan, H. Y .,Proc. I .R.E . 40, 1342 (1952); Haynes, J. R., and Westphal, W., Phys. Rev. 86,680 (1952); Haynes, J. R., and Hornbeck, J. A., Phys. Rev. 90, 152 (1953). 189. Fan, H. Y., Navon, D., and Gebbie, H. A., Physica 20,855 (1954). 130. Tyler, W. W., Bull. Am. Phys. SOC.30, No. 3 (DA3), 20 (1955). 191. Haynes, J. R., and Hornbeck, J. A., “The Trapping of Minority Carriers in Silicon,” presented a t the AtIantic City Conference on Photoconductivity (1954) ; Hornbeck, J. A., and Haynes, J . R., Phys. Rev. 07, 311 (1955). 136. Van Roosbroeck, W., and Shockley, W., Phys. Rev. 04, 1558 (1954). 139. Burstein, E., Picus, G., and Teitler, S., unpublished. 13.4. Haynes, J. R., and Briggs, H. B., Phys. Rev. 86, 647 (1952); Haynes, J. R., Bull. Am. Phys. SOC.80, No. 4 (U7) 30 (1955). 136. Aigrain, P., Physica 20, 1010 (1954). 136. Galkin, L. N., and Korolev, N. V., Doklady Akad. Nauk S.S.S.R. 92,529 (1953); Garlick, G. F. J., and Dumbleton, M. J., Proc. Phys. Soc. (London) B67, 442 (1954). 197. Avery, D. G., Goodwin, D. W., Lawson, W. D., and Moss, T. S., Proc. Phys. Soc. (London) B67, 761 (1954). 138. Talley, R. M., and Enright, D. P., Phys. Rev. 96, 1092 (1954). 199. Taft, E. A., and Hebb, M. H., J . Opt. SOC.Amer. 42, 249 (1952); Bube, R. H., “ Superlinear Photoconductivity,” presented at Atlantic City Conference on Photoconductivity (1954); Rose, A., “ Performance of Photoconductors ” presented a t Atlantic City Conference on Photoconductivity (1954).

PHYSICS OF SEMICONDUCTOR MATERIALS

83

140. Haynes, J. R., and Shockley, W., Phys. Rev. 81, 835 (1950); Prince, M., Phys. Rev. 92,681 (1953); 93, 1204 (1954). 141. Willardson, R. K., Harman, T. C., and Beer, A. C., Phys. Rev. 96, 1512 (1954). 142. See discussion by Lark-Horovita, K., i n “The Present State of Physics,” p. 57. American Association for Advancement of Science, Washington, D.C., 1954. 143. Seitz, F., Phys. Rev. 79, 372 (1950); Estermann, I., and Foner, A., ibid. 79, 365 (1950). 144. Macfarlane, G. G., and Roberts, V., Phys. Rev. 97, 1714 (1955). 146. Madelung, O., and Weiss, H., 2. Naturforsch. 9a,527 (1954). 146. Avery, D. G., Proc. Phys. SOC.B66, 425 (1953); B66, 134 (1953). 147. Moss, T. S., Proc. Phys. SOC.B67, 775 (1954). 148. Kittel, C., Am. J. Phys. 22, 250 (1954). 149. Herring, C.,Phys. Rev. 96, 1163 (1954). 160. Dresselhaus, G., Kip, A. F., Kittel, C., and Wagoner, G., Phys. Rev. 98, 556 (1955). 161. Geballe, T., and Hull, G. W., Phys. Rev. 94, 1134 (1954). 161. Barrie, R., Cunnell, F. A., Edmond, J. T., and Ross, I. M., Physica 20, 1087 (1954). 166. Putley, E. H., Proc. Phys. SOC.(London) B68, 35 (1955). 164. Benedict, T. S., and Shockley, W., Phys. Rev. 80, 1152 (1953); Benedict, T. S., ibid. 91, 1565 (1963); Goldey, J. M., and Brown, S. C . , Bull. Am. Phys. SOC.30, No. 1 (V3), 53 (1955). 166. Dexter, R. N., and Lax, B., Bull. Am. Phys. SOC.30, No. 3 (L2), 35 (1955). 166. Robertson, R., Fox, J. J., and Martin, A. E., Phil. Trans. Roy. SOC.A232, 463 (1934). 167. Willardson, R. K., and Danielson, G. C., J. Opt. SOC.Amer. 42, 42 (1952). 168. Klick, C. G., and Maurer, R., Phys. Rev. 81, 124 (1951). 169. Redfield, A. G., Phys. Rev. 94,526 (1954). 160. Morin, F. J., and Maita, J. P., Phys. Rev. 94, 1525 (1954). 161. Prince, M. B.,Phys. Rev. 92,681 (1953). 16.6’. Morin, F. J., and Maita, J. P.,Phys. Rev. 96, 28 (1954). 163. Ewald, A. W., Phys. Rev. 91, 244 (1953); Becker, J. H., Bull. Am. Phys. SOC.SO, No. 1 (V6) (1955). 164. Ewald, A. W., J. Appl. Phys. 26, 1436 (1954). 166. Busch, G., Wieland, J., and Zoller, H., Helv. Phys. Acta 23, 528 (1950). 166. Busch, G., and Mooser, E., Helv. Phys. Acta 26, G l l (1953). 167. Kendall, J. T., Phil. Mag. [7] 46, 141 (1954). 168. Ewald, A. W., Phys. Rev. 04, 1428 (1954); 97,607 (1955). 169. Becker, J., private communication. 170. Bolton, V. E., Science 116, 571 (1952); Fukuroi, T., Tanuma, S., and Tobisawa, S., Science Repts. Research Insts., T6hoku Univ. ser. A4, 283 (1952). 171. Loferski, J. J., Phys. Rev. 93,707 (1954). 17.6’. Moss, T. S., Proc. Phys. SOC.(London) 62, 590 (1951). 178. Bardeen, J., Phys. Rev. 76, 1777 (1949); Bridgeman, P. W., Proc. Am. Acad. Arts Sci. 72, 159 (1938); 77, 187 (1949). 174. Neuringer, L. J., Bull. Am. Phys. SOC.30, No. 1 (U9) 54 (1955). 176. Middleton, A. E., unpublished. 176. Fukuroi, T., and Tanuma, S., Science Repta. Research Insts. TMoku Univ. aer. A4, 353 (1952).

84 177. 178. 179. 180. 181. 182,

186. 284. 186. 186. 187. 188. 189. 190. 191.

ELIAS BURSTEIN AND PAUL H. EGLI

Fritsche, H., Science 116, 571 (1952). Callen, H., J . Chem. Phya. 22, 518 (1954). Nussbaum, A., Phys. Rev. 94, 337 (1954). See review by Smith, R. A., Physica 20, 910 (1954). Scanlon, W. W., Phys. Rev. 92, 1573 (1953); Physica 20, 1090 (1954). Tanenbaum, M., and Briggs, H. B., Phys. Rev. 91, 1561 (1953); see also Austin, I. G., and McClymont, D. R., Physics 20, 1077 (1954); Breckenridge, R. G., Blunt, R. F., Hosler, W. R., Frederikse, H. P. R., Becker, J. H., and Oshinsky, W., Phys. Rev. 96, 571 (1954). Tanenbaum, M., and Maita, J. P., Phys. Rev. 91, 1009 (1953). Hrostowski, H. J., Wheatley, G. H., Flood, W. F., Jr., Phys. Rev. 96,1683 (1954). Burstein, E., Phys. Rev. 93, 632 (1954). Folberth, 0. G., Grimm, R., and Weiss, H., 2. Naturforsch. 8a, 826 (1953); Folberth, 0. G., Madelung, O., and Weiss, H., 2.Naturforsch. 9% 954 (1954). Blunt, R. F., Hosler, W. R., and Frederikse, H. P. R., Phys. Rev. 96,576 (1954). Oswald, F., Z. Naturforsch. 9a, 181 (1954); Oswald, F., and Schade, R., 2. Naturforsch. Qa, 611 (1954); Cunnell, F. A., Edmond, J. T., and Richards, J. L., Proc. Phys. SOC.(London) B67, 848 (1954). Folberth, F., and Oswald, F., Z.Naturforsch. 9a, 1050 (1954). Keck, P. H., and Wolf, O., private communication. Busch, G., and Winkler, V., Helv. Phys. Aeta 26, 395 and 578 (1953); Physica 20,

(1954). 19.9. Blunt, R. F., Frederikse, H. P. R., and Hosler, W. R., Bull. Am. Phys. SOC.30, No. 2 (D7), 10 (1955). 193. Braunstein, R., Phys. Rev. 99, 1892 (1955). 194. Frederikse, H. P. R., and Mielcsarek, E. V., Phys. Rev. 99, 1889 (1955).