Chapter 1 Band Structure and Galvanomagnetic Effects in III-V Compounds with Indirect Band Gaps*

CHAPTER 1

Band Structure and Galvanomagnetic Effects in 111-V Compounds with Indirect Band Gaps* Richard J . Stirn I. INTRODUCTION . . . . . . . . . . . . . . . 11. RELATION OF GALVANOMAGNETIC EFFECTS TO BANDSTRUCTURE. 1. Conduction Band. . . . . . . . . . . . . . 2 . Valence Band . . . . . . . . . . . . . . 111. EXPERIMENTAL RESULTS . . . . . . . . . . . . 3. Aluminum Antimonide . . . . . . . . . . . . 4. Gallium Phosphide . . . . . . . . . . . . . 5. Aluminum Arsenide . . . . . . . . . . . . 6 . Aluminum Phosphide . . . . . . . . . . . . I . Aluminum Nitride . . . . . . . . . . . . . 8 . Boron Nitride . . . . . . . . . . . . . . 9. Boron Phosphide . . . . . . . . . . . . .

. . .

. .

1

3 3 14

. .

22 22 38

. .

58 61 62

. .

.

54

64

I. Introduction The subject material in earlier volumes of this series has been predominantly concerned with 111-V compounds with a “direct band gap.” These compounds have both the valence-band maxima and the lowest conduction band minima located at or near’ the center of the Brillouin zone (I-‘). In this chapter, attention will be fixed upon those 111-V compounds which do not have their conduction-band minima located at k = 0. Instead these minima, which are ellipsoids of revolution in momentum space, are apparently located along the [100l-axes (A). These compounds, as well as the direct-gap compounds, appear to have nearly the same type of valence band structure.2,2aThis structure is very similar to that found in germanium and silicon,’ and will be described in some detail below.

* This chapter was prepared at the Jet Propulsion Laboratory, a center operated by the California Institute of Technology with the support of the National Aeronautics and Space Administration. The siight lifting of the valence-band degeneracy near k = 0 for the upper two valence bands in compounds with the zinc-blende lattice will not be of concern in this chapter. R. Braunstein and E. 0.Kane, J . Phys. Chem. Solids 23, 1423 (1962). ’*F. Bassani and M. Yoshimine, Phys. Rev. 130,20(1963). 1

2

RICHARD J. STIRN

Theories relating band structure to galvanomagnetic effects for minima of ellipsoidal symmetry and warped maxima of near spherical symmetry are reviewed in Part 11. The experimental data available for the indirect gap compounds are presented in Part 111, with emphasis on more recent results. Most of the evidence for the band structure in these compounds comes from recent theoretical and optical studies. Since these studies have not been presented in detail in other reviews, pertinent results from them have also been included. High carrier mobilities and homogeneous single crystals of high purity are important in the type of experiments which can give band parameters directly, such as cyclotron resonance and magnetoresistance. Single crystals of the indirect-gap 111-V compounds are generally more difficult to grow with fewer impurities than is possible with most of the direct gap compounds. This is due to the fact that the former have much higher melting point temperatures, and hence, have considerably more chemical reaction with the surroundings. Three of the indirect-gap compounds, AlAs, AlP, and AlSb are unstable as crystals when left exposed to moist atmosphere. In addition, electron mobilities in the indirect-gap compounds are substantially lower than those of the direct-gap compounds, due in part, to the fact that carriers in minima away from the zone center have higher effective masses than carriers in the central minimum. For such reasons as these, experimental measurements of the transport properties are comparatively rare for the compounds of interest in this chapter. Most of the knowledge about effective masses has been obtained from optical and magnetooptical measurements. Values of the energy gaps and assignment of the various minima in k space have also been derived, for the most part, by optical absorption and reflectance studies and their pressure dependence. Despite certain fabrication difficulties, there is great interest in the indirectgap 111-V compounds from a device standpoint. Their large band gaps, and consequently, their ability to operate at higher ambient temperatures is one reason for this interest. As an example, G a P has shown some promise as a power rectifier at temperatures as high as 500°C.3 Higher photovoltages can be obtained with semiconductors having larger band gaps. However, the ability to generate electron-hole pairs is fixed by the gap energy and the spectral distribution of the light to be used. In the case of sunlight in the absence of an atmosphere, an ideal solar cell using material with a gap energy (E,) of 1.5--1.6eV would yield the highest theoretical R. E. Davis, Metallurgical Society Conference, “Properties of Elemental and Compound Semiconductors” Vol. 5, p. 295. Wiley (Interscience), New York, 1960.

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

3

efficiency (y z 24%).4 Thus AlSb would appear to be the most likely candidate for a solar battery operating in outer space, if one does not consider fabrication or purity problems. In practice, because of these problems and the effect of the atmosphere, which shifts the maximum of the q - E, curve toward lower gap energies, more attention is being devoted to the compounds GaAs and InP. The field of electrooptical modulation5 is currently of great interest, since electrooptical crystals can be used to modulate the intensity of light beams, shift the beam’s position, and vary the pass band of a filter element. GaP could be useful as an electrooptical modulator since it is transparent to a large portion of the long wavelength region of the visible spectrum. It is cubic and therefore would have fewer alignment problems. It has a relatively low dielectric constant, which reduces capacitance effects. Finally, GaP can be used in the transverse electric-field mode, which allows a longer path length to be used without increasing the applied voltage excessively. Stimulated light emission is not expected to be exhibited in the indirectgap compounds because of their small radiative recombination cross section for band-to-band transitions. However, alloying a compound that does give stimulated light emission with the appropriate indirect-gap compound enables the frequency of the emission to be shifted to a higher frequency. For example, the alloy (GaAs, - xPx),in which this shift was first observed by Holonyak and Bevacqua,6 is receiving much attention. The growth and preparation of single-crystal, indirect-gap 111-V compounds has been reviewed elsewhere.’ Some references to more recent improved growth techniques for GaP are given in the section on Gap. 11. Relation of Galvanomagnetic Effects to Band Structure

1. CONDUCTION BAND Conduction electrons in a solid subjected to a magnetic field H while an electric field E is present experience a Lorentz force. The manner in which they react is much different from the corresponding situation in a vacuum. In a solid, the nature of the scattering mechanisms and the anisotropy of the energy minima in momentum space modify the electrical current. This J. J. Loferski, .I Appl. . Phys. 27,777 (1956). For good review articles see, e.g., I. P. Kaminow and E. H. Turner, Appl. Opt. 5, 1612 (1966); A. R. Johnston, AGARD Optoelectron. Components Conj p. 129. Technical Editing and Reproduction Ltd. London, 1965. N. Holonyak and S. F. Bevacqua, Appl. Phys. Lett. 1, 82 (1962). ’ R. K. Willardson and H. L. Goering, eds., “Preparation of 111-V Compounds.” Reinhold, New York, 1962.

4

RICHARD J. STIRN

electrical current can be described in a completely general form in tensor notation as j i =,oijEj

+ o i j l E j H ,+ oi,l,EjHlH,.

(1)

In writing Eq. (I), we have explicitly assumed weak magnetic fields by retaining only terms to second order in magnetic field. The coefficients defined in Eq. (1) are elements of a generalized conductivity tensor. An element of the second rank conductivity tensor oij = (dj,/dE,),=,

describes the current in the absence of a magnetic field. In a cubic system, , ~ aij is the Kronecker delta, this conductivity is a scalar, i.e., oij = 0 ~ 8where equal to zero when i # j and equal to one otherwise. An element of the third rank conductivity tensor

a ~ ,

oijl = a 2 j i / a E j

is associated with the Hall effect. It is also nondirectional in a cubic system, i.e., aijl = ao2R,eij,, where R, is the Hall coefficient in the limit of zero magnetic field and eijIis the permutation tensor, defined in the usual manner, e I 2 3 = E~~~ = E~~~ = 1, e Z l 3 = ~ 1 3 2= ~ 3 2 1= -1, with all other components being zero. The last coefficient, an element of the fourth rank conductivity tensor

= a 3 j i / 2 !aE, a H , a H , gives the weak-field magnetoresistance, or more properly, magnetoconductivity. Now, however, the values of aijlmdepend upon the directions ijlrn even for a cubic system. Even more important, the values are also dependent on the system of conduction minima present in the material. It is for this reason that directional magnetoresistance measurements can provide us with information as to the direction of the minima in k space. The coefficients in Eq. (1) can be expressed as transport integrals, which can be solved in a closed form under certain conditions. Abeks and Meiboom' and Shibuya' obtained expressions for the coefficients for the case of a scalar relaxation time and an arrangement of the minima consisting of either : (a) 3 or 6 ellipsoids of revolution (ml* = rn2* # m 3 * ) with major axes along the [loo] directions, or (b) 4 or 8 ellipsoids of revolution with major axes along the [ l l l ] directions. The case of [110] ellipsoids of revolution was also treated by Shibuya' and that of [110] ellipsoids with all three B. Abeles and S. Meiboom, Phys. Rev. 95, 31 (1954). M. Shibuya, Phys. Reo. 95, 1385 (1954).

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

5

'

effective masses unequal by Allgaier. l o Later, Herring and Vogt extended the results to include an anisotropic relaxation time when z has the same symmetry as the energy ellipsoids. Their treatment is applicable to scattering processes which either conserve energy or randomize velocity. These include intervalley and deformation-potential intravalley scattering, which are usually the dominant scattering mechanisms in indirect 111-V compounds. The assumption is not good for polar mode, piezoelectric, and charged center scattering, which in practice, play a minor role. In the cases where the Herring-Vogt anisotropic-relaxation time treatment is valid, the transport properties are described by expressions identical with those for isotropic scattering, except that each component of the reciprocal effective mass tensor is weighted by. the corresponding relaxation time component. Thus, the combinations z , / m l * = zz/m2* # z3/m3*always occur. A parameter that will describe the anistropy in an ellipsoid of revolution (valley) is the effective-mass anisotropy parameter K = (m */m,*)(zl/z 11), where we have chosen the indices such that m,* = m,* = m,* and m3* = mll. The subscripts II and I denote directions parallel and perpendicular, respectively, to the axes of revolution of the ellipsoids. When one sums over all valleys, the parameter K can be used in the final expressions" for the coefficients in Eq. (l), when zJl and zl have the same energy dependence. These expressions are presented next. a. Electrical Conductivity

The carrier concentration in the (nondegenerate) rth valley is

The contribution of the rth valley to the electrical conductivity, in the principal-axis system of the valley, is given by

where rm

lo

I om

R. S. Allgaier, Phys. Rev. 115, 1185 (1959). C. Herring and E. Vogt, Phys. Reu. 101,944 (1956).

6

RICHARD J. STIRN

fo being the zero-field Fermi distribution function. The relaxation time is assumed to have the form z = TOES.

(3b)

The total contribution is then obtained by summing over all valleys : 60 =

w ( ( q > / m l l * ) ( 2 K+ 11,

(4)

where n is the total carrier concentration. The expression for the electron conductivity mobility (at zero magnetic field) follows immediately from Eq. (4) as Po

=

-fe((zII>/ml/*)(2K + 1).

(5)

b. Hall EfSect As was the case for go, the Hall constant is independent of direction in a cubic crystal. The contribution of the rth valley, in the principal-axis system of the valley, is

Summing over all axially symmetric valleys yields

The zero-field Hall mobility is defined as poH= Roooc, where c is the velocity of light.12 Then aijl= O , ~ R , E ,=~ ~o o ( p o H / ~ ) ~ i j l .

(8)

It follows from Eqs. (4), (9,(7), and (8) that

The fundamental quantity r = Ro/R, = - nec R,, in Eq. (9), where R , is the Hall constant at infinite magnetic field, is of particular interest. Averaging over the relaxation time in Eq. (9) by use of Eq. (3) gives (z112)/(z1,)2 = 3x/8 = 1.18 for acoustical mode scattering (s = -+), 315n/512 = 1.94 for ionized impurity scattering and electron-hole scattering (s = *), and 1.00 for neutral impurity scattering (s = 0). For the case of polar mode scattering, the value of (zll 2)/(z11)2as a function of temperature has When lab units are used, c is replaced by unity. The units of p H , R , , and uo are then (cm' V - ' sec-I), (cm3coul-'), and (ohm-cm)-', respectively.

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

1.25

I . / < ~ > 2 -1.

A

fi

7

-1.27

I8

I

FIG.1. Relaxation-time averages ( T ’ > / ( T ) ~ and ( T ~ ) ( T ) / ( T ’ ) ’ as a function of the ratio 6pJp, for admixtures of acoustic phonon and ionized impurity scattering.

=

been calculated by variational techniques in the weak-field region.’3’13a,13b The value at a given temperature depends upon the characteristic temperature 8, associated with the longitudinal optical frequency and the amount of degeneracy, but never becomes greater than about 1.3. Stillman et ~ 1 . l have compared their experimental results for r versus temperature with the theory 13*13dfor the case of polar scattering in GaAs ( K = 1). In actual practice, of course, the true value of ( r 1 1 2 ) / ( ~ 1will 1)2 be determined by a combination of scattering mechanisms. Since the mobility due to polar scattering is nearly exponential in temperature dependence, it would be a reasonably good approximation to consider only ionized impurity and acoustical scattering at temperatures below about 8J2. By adding the reciprocal relaxation times and dropping the slowly varying logarithmic factor in the ionized impurity relaxation time, one can obtain a total relaxation time l4

B. F. Lewis and E. H. Sondheimer, Proc. Roy. SOC.(London)A227,241 (1955). I3”H. Ehrenreich, J. Phys. Chem. Solids 8, 130 (1959); 9, 129 (1959). 13bD.J. Olechna and H. Ehrenreich, J . Phys. Chem. Solids 23, 1513 (1962). 13cC. E. Stillman, C. M. Wolfe, and J. 0. Dimmock, J. Phys. Chem. Solids 31, 1199 (1970). 3dS.S. Devlin, in “Physics and Chemistry of 11-VI Compounds” (M. Aven and J. S. Prener, eds.), p. 561. North-Holland Publ., Amsterdam and Wiley (Interscience), New York, 1967. l4 See,e.g., A. C. Beer, Solid State Phys. Suppl. 4, 145 (1963). l3

~

~

8

RICHARD J. STIRN

where pL is the acoustical lattice mobility, pI the mobility due to ionized impurity scattering, x = e/kT, and p = 6pL/p1is a measure of the amount of ionized impurity scattering. The results of calculating ( T ~ ~ ~ ) / ( T as , , )a~ function of p are given in Fig. 1. Since the contribution of the band structure in Eq. (9) lowers r by 13-21 % (for K values from 5 to 20) the usual (and sometimes inappropriate) approximation r = 1 is actually quite reasonable for many-valley semiconductors. c. Weak Field Magnetoresistance The contribution of the rth valley to the magnetoconductivity, in the principal-axis system of the valley, is

As was mentioned earlier, the total contribution from all valleys depends upon the arrangement of valleys in the Brillouin zone. Values for the [loo] and [l 111 systems have been given by Herring and Vogt" and are reproduced in Table I, which is to be used in conjunction with the following expression obtained by summing over all axially symmetric valleys :

The coefficients g 3 0 , g , for each set of valleys.

and g , , are the tabulated quantities given in Table I TABLE I

LOW-FIELD MAGNETOCONDUCTIVITY COMPONENTS FOR CUBIC CRYSTALS WITH AXIALLY SYMMETRIC VALLEYS REFERRED TO THE CRYSTAL AXES Type of valleys (direction of k) Component, referred to crystal axesh

[1001

[1111

From Herring and Vogt." The quantities b, c, and d are defined in Eqs. (16), (20H27). Tabulated quantities are the coefficients g,,, g2,, and g , , , respectively, in Eq. (12).

Direct application of Eq. (12) to experiment, i.e., prediction of the magnitude of the magnetoresistance effect, is usually not feasible because of the

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

9

lack of knowledge about the magnitude and energy dependence of z. This is particularly true for the III-V compounds, for which a relaxation time is not even definable in the temperature range where polar scattering is dominant.' A phenomenological theory applicable to cubic crystals, which was first given by Seitz,15 proves to be a very convenient way of analyzing magnetoresistance. Seitz expressed the current density in an equation that included second-order terms in magnetic field strength as j

=

aoE

+ cc(E x H) + / W E + y(E - H)H + 6TE,

(13)

where Tis a diagonal tensor with elements H12,H Z 2 ,and H , 2 (the coordinate axes 42, and 3 being taken along the crystal axes). The coefficient a, which is related to the Hall constant, and the magnetoresistance coefficients B, y, and 6 were given by Seitz in terms of transport integrals. These coefficients and the zero-field conductivity oo are related to the basic conductivity components of Eq. (1) as follows:

Experimentally the current is kept constant rather than the electric field, thus the inverted form of the Seitz equation16 is more useful for analysis. To second order in H ,

E

=

po[j

+ a(j x H) + bH2j + c(j.H)H + dTj],

(15)

where PO

a

= 1/00,

c

= -ape,

= -(y -

poct2)po,

b= d

-(P + P O " ~ ) P O ,

=

-6p

0.

(16)

The coefficients b, c, and d are known as the inverted Seitz coefficients. The magnetoresistance follows immediately from Eq. (15) : M$

E

( [ p - p0]/p0H2)f,","= (E - E,=,). j/(E,=, . j ) H z

=

b

+ c[Q. H)2/j2H2]+ d(j12H12+ jz2HZ2+ j32H32)/j2H2,

which can be rewritten as

M!?

=

b

+ ~(Zi;lq)~ + dCiZ;lq2,

(17)

(18)

where L and q are the direction cosines of the current and magnetic-field vectors with respect to the cubic axes. The sub- and super-indices of the l5

l6

F. Seitz, Phys. Rev. 79, 372 (1950). G . L.Pearson and H.Suhl, Phys. Rev. 83,768 (1951).

10

RICHARD J. STIRN

magnetoresistance coefficient, M , indicate the directions of the current and magnetic-field vectors, respectively. In the case of isotropic materials ( K = 11, b+c=O, d=O. (19)

In the case of many-valley semiconductors, expressions for b, c, and d can be obtained by manipulating Eqs. (4),(5), (9), (12) and using the information given in Table I. These expressions are dependent only upon the anisotropy parameter K , the zero-field Hall mobility poH,and the dimensionless quantity A

= (23)(t)/(22)2.'7

The results are12 : System of [ I O ~ l - t y p espheroids'

b

=

7a

+ + 1)(2K + 1)/K(K + 2)']

( ~ O " / C ) ~ { A [ ( KK'

-

l},

(20)

System of [ 11 11-type spheroids'

+

+ 2)] - l } ,

b

=

(P,~"/C)~{A[(~K 1)'/3K(K

c

=

-(po"/~)~{A[(2K 1)2/3K(K 2)] - I ) ,

+

d = (pO"/c)'{A[2(K- 1)'(2K b+c=O,

+

+ 1)/3K(K + 2)2]}, d>O.

(24) (25) (26) (27)

The symmetry relations Eqs. (19), (23), and (27) are independent of the particular form of the relaxation time. Values of the inverted Seitz coefficients can be obtained experimentally by a minimum of three magnetoresistance measurements with the current and magnetic field along certain high-symmetry directions. From Eq. (18), it may replace 711with T in the quantity A, ifwe maintain the assumption that both z,, and zL have the same energy dependence. ""Note that the Seitz coefficient c is not to be confused with the speed of light c. The latter occurs in the ratio (ploH/c). " We

1. follows that

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

+ d/2, Moo‘ - M0’O - b Milo 11’ - MI1’ 1 1 1 = b + d/3, Mf:; = b + d/6, Mi:: = b

110

100

11

7

+ c + 4 2 , Mloo = b + c + d, Mill = b + c + d/3.

Ml12 = M l l o = b

(28)

The absence of a superscript implies that the current and magnetic field are collinear. The longitudinal magnetoresistance vanishes when the magnetic field is along the cubic axes of a crystal having [loo]-type minima. The longitudinal effect is a maximum when the field is along a (1 11) direction for the [loo] system of spheroids, and when the field is in a (100) direction for the [ l l l ] system. The values of b, c, and d enable one to determine the anisotropy of a valley, in addition to identifying the system of valleys by use of the symmetry relations. The energy dependence of the relaxation time, i.e., A, can be eliminated from Eqs. (20H22) or Eqs. (24x26) by solving two of the equations from each set simultaneously. A convenient parameter to use, which requires only two transverse measurements, is q E [b (pOH/c)’]/d. We then obtain

+

+ 3K/(K - 1)’ [lo0 valleys], (K,)ioo = 3q - 1)-’[(2q + 1) f (12q - 3)”’1, q l l l = 1 + 9K/2(K - 1)2 [111 valleys], q l o o= 1

and

(29) (30) (31)

The parameter q is plotted in Fig. 2 for both systems as a function of K , and I(-, where K , (> 1) corresponds to the value of the effective mass-relaxation-time anisotropy for prolate spheroids (m > m,) and K - (< 1) is the value for oblate spheroids (ml, < ml). It is interesting to note that the value of K , is relatively insensitive to errors in b, d or the Hall mobility for the case of semiconductors with low anisotropy ( K 2 5). However, when the anisotropy is high, careful measurements and the elimination of spurious magnetoresistance effects due to contact shorting’ ‘*19 and the presence of various types of inhomogeneities20*20a are required in order to obtain a reasonable certainty in the value of K + . J. R. Drabble and R. Wolfe, J. Electron. Conlrol3, 259 (1957). R. F. Broom, Proc. Phys. SOC.(London)71,500 (1958). 2 o A. C. Beer,14 p. 308. 20aR. T. Bate, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 4, p. 459. Academic Press, New York, 1968. l9

12

RICHARD J . STIRN

FIG.2. Anisotropy parameters K + and K - versus q.

The choice between the prolate and oblate spheroidal model rests upon the magnitude of A, which can now be calculated by substituting the value of K + or K - into any one of the equations for b, c, or d for the appropriate system of valleys. The parameter A cannot be less than 1, since (t3) ( t ) is greater than or equal to by Schwarz’s ineqrlality. Values of A are given in Fig. 1 for a range of admixtures of acoustic phonon and ionized impurity scattering. d. Effective Masses

Because of interaction between the electron and the crystal lattice, the electron effective mass m* is different from the free-electron mass m,. In addition, when the energy surfaces are ellipsoids of revolution, the effective mass of the electron varies, depending upon its position in momentum space. In general,

(m*- l)ij if there is no degeneracy.

=

(l/ti2) d2&(k)/dkid k j

(33)

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

13

The coordinate system is usually chosen so that the reciprocal mass tensor of Eq. (33) is reduced to principal axes. In this system, the quadratic relationship between energy and wave number becomes

E(k) = ~h’[k,’/m,*

+ k2’/m2* + k 3 2 / m 3 * ] ,

(34)

where we have chosen the energy zero to be at the minimum. Because of differences in the method of averaging the mass components, it is important to distinguish between several types of effective masses in the case of multi-ellipsoidal bands. Thus, by comparing Eq. (4) with the usual expression for the zero-field conductivity, oo = ne2z/mo, a conductivity effective mass mc* is defined as

(me*)-’

=

gl/m,*

+ l/m2* + l/m3*) = ( 2 K + 1)/3mll*.

(35)

This mass is used in relationships concerning infrared reflectivity and free carrier absorption, for example. It is conceptually useful to define a parameter which is a measure of the density of states available in a given energy range irrespective of the shape of the energy surface. The density of states is proportional to the volume of phase space, so that if we replace the volume of the sphere of radius k/(2m*)’/’ used for spherical energy bands with the corresponding volume of an ellipsoid with semimajor axes dk1/(2m1)’/’,dk2/(2m2)’/2,and dk3/(2m3)’/’,the h a 1 calculation will include a density-of-states effective mass md, where for each valley md = ( m , m, m3)’i3. (36) If v is the total number of equivalent valleys, then the expression for the total carrier density will be of the form n a v(ml m2 m3)’l2 a md*3’2, where md* here refers to all of the equivalent valleys. The density-of-states effective mass for the multivalley system is then given by

md* = ~ ~ / ~ ( m3)’/3 m , m=~v ~ / ~I*. K ~ / ~ (364 ~ This mass is applicable to thermoelectric-power theory and is required when relating the carrier concentration to the Fermi level. An additional type of effective mass is one that appears in high-frequency Faraday rotation theory for many-valley semiconductors.” If the amount of rotation per unit length per unit magnetic field is expressed using the zerofield Hall coefficient rather than the free-carrier concentration, a Faraday effective mass can be defined as mF* = (2K l)m,*/(K 2). (37)

+

+

M. J. Stephen and A. B. Lidiard, J . Phys. Chem. Solids 9,43 (1958).

14

RICHARD J. STIRN

A reliable determination of any two of these three effective masses would uniquely give the two mass components m,,* and m,*. Alternatively, a value of just one of the effectivemasses would be sufficient if the anisotropy K can be determined from cyclotron resonance or magnetoresistance measurements. Requirements of crystal purity are less stringent for the latter type of measurement. However, an additional uncertainty in the value of K is introduced by anisotropy of the relaxation time. 2. VALENCE BAND In presenting the effect of conduction-band structure on galvanomagnetic phenomena, we had need only to consider single-band conduction for the compounds of interest here. This is because of their large band gaps ( 21.6 eV) and the fact that higher-lying bands appear to be at least 0.30 eV above the [loo] minima (Part 111). However, in p-type material, the presence of multiband conduction greatly affects galvanomagnetic phenomena, complicating the interpretation of experimental data. This multiband conduction occurs in the 111-V compounds, as it does in germanium and silicon, because of the presence of two bands degenerate (excluding spin) at k = O’, having very different curvature : Vz-band (heavy holes) and V3-band (light In addition, there is a third band lying lower in energy, split off from the other two because of spin-orbit interaction. In the 111-V compounds, however, the split-off band is thought to be far enough removed in energy so that its contribution to the conductivity is negligible. An additional factor that makes the interpretation of galvanomagnetic properties for p-type material much more complicated than for n-type material is the fact that the bands V, and V3 are warped from spherical symmetry for some distance away from the zone center.23It is for this reason that closed expressions for the galvanomagnetic coefficients oijl and oijlm are not possible, unlike the case for minima of spheroidal symmetry. Instead, the expressions must include the anisotropy in the form of a rapidly converging infinite series. For degenerate valence bands, the carrier energy dependence on the wave vector k was first given by Dresselhaus et aLZ3for germanium and silicon by the function E(k)

1

-(h2/2m,){Ak2 ’. [B2k4

+ C2(kx2ky2+ kyzkz2 + kr2kx2)]”2},

(38)

for a k-coordinate system along the cube axes. A, B, and C are warping parameters which determine the curvature and deviation from sphericity of the energy bands away from k = 0. This expression is applicable to 111-V 22

23

F. Herman and J. Callaway, Phys. Reu. 89, 518 (1953). G. Dresselhaus, A. Kip, and C. Kittel, Phys. Rev. 95, 368 (1954).

1.

15

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

compounds if we neglect the very small effect of the linear terms in k near the zone center, which is due to the lack of inversion symmetry.2 Lax and Mavroides have described a method for carrying out calculations which deal with problems related to the hole densities, conductivities, effective masses, and Hall coefficient^,'^ as well as magnetoconductivities.Z The calculations are valid for weak magnetic fields and assume an isotropic relaxation time. The key to these calculations was the rewriting of Eq. (38) so that the expressions under the radical were expanded in powers of an anisotropy term, in such a way that the series converged rapidly for typical degrees of warping. Thus E(k) = - (h2k2/2mo)(Af B’){1 - T[(k:ky2 ky2k? + k,2k,)/k4] * * * } ,

+

+

where

r = T C ~ / [ ~ B ’ (f A B’)],

B’ = (B2 + C2/6)1’2.

(41)

The upper sign in Eqs. (38H40) and in forthcoming expressions is associated with the holes of smaller effective mass. Equations (38) and (39) reduce to the usual quadratic dependence of energy upon k when the amount of warping becomes vanishingly small (C -+ 0). In this case, the effective masses are given by m* = ( A f B)-’mo. The wave-vector dependence in Eq. (39) was then represented in spherical coordinate^,'^ after which integrations could be performed. The resulting expressions for the hole density, conductivity, Hall coefficient, and magnetoconductivity coefficients, which are presented in the following subsections should be applicable to all of the 111-V compounds.

a. Electrical Conductivity and Hole Density The number of holes per unit volume is given by (42)

P n = Ps(ad)n?

for a Maxwellian distribution, where ps

=

2[2.nrn0k T / ~ ’ ( A B’)]3/2e(E~-EF)/kT

(43)

is the carrier concentration for a spherical energy surface, k is the Boltzmann constant, and E , is the valence-band energy at k = 0. The anisotropy part is

+

(ad)n= 1 + o.05rn + o.oi635rn2 o.ooo908rn3

where n 24

=

+ ...,

2, 3 denotes the heavy- and light-hole bands, respectively.

B. Lax and J. G. Mavroides, Phys. Reo. 100, 1650 (1955).

*’ J. G. Mavroides and B. Lax, Phys. Rev. 107, 1530 (1957);108, 1648 (1957)

(44)

16

RICHARD J . STIRN

The ratio of light- to heavy-hole carrier concentration is then given by P3/P2 =

(LA

-

B'l/[A + B'l)3'2(ad)3/(ad)2.

(45)

The electrical conductivity in either band can be written as

where all

=

+ 0.016671- + o.04i369r2 + o.ooo90679r3 + o.ooo91959r4 + 0.0000210~r5+ . . . ,

1

(48)

and the zero-field conductivity mobility for a spherical energy band is pOs = Iel(z)(A f B')/rno

(B' -+ B ) .

(49)

From Eqs. (43) and (47), it is seen that the ratio of the zero-field conductivity mobility in a warped band, p o , to that in a spherical band is PO/POs =

(50)

l/ad.

The total conductivity oo, which is isotropic, is simply the sum of the individual conductivities : 00

='Iel[{PSPo"a,1)2

+ cPsPos~lls31.

(51)

Thus, the total drift mobility, p o , is Po

=

[(P& + PdP2(110)31/[1 + P3/P219

(52)

if we neglect the anisotropy parameters a l l and ad. A useful quantity is the light- to heavy-hole lattice mobility ratio since it is a measure of the effect of light holes on galvanomagnetic phenomena. This ratio can be written as &,3/&,,2

=

( A + B')/(A -

B')3

(53)

if the relaxation times of both holes are equal. The scattering rate of a carrier is proportional to the density of final states. If the scattering of the light hole is predominantly interband, as it is in Ge,26 then the assumption of equal relaxation times should hold since the final state would be the same for both the light and heavy holes, namely, the heavy-hole band. 26

See, e.g., H. Brooks, Aduan. Electron. Electron Phys. 7, 152 (1955)

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

17

b. Hall Coeficient The coefficient for the Hall term in Eq. (1) for warped degenerate bands has been given by Lax and MavroidesZ4and can be written for either band as oijt =

(psle31/mo2c)(z2>(A~t W’aiZ~iji,

= PsIeI~os(~~,s/~)a12&ij~r

where

+ 0.0179561-2 -o.oo69857r3 + 0.0012610r4 + . . .

a 1 2 = 1 - 0.016671-

(54) (55)

(56)

and p!,s is the Hall mobility at zero-magnetic field for a spherical energy surface. The weak-field limit of the Hall coefficient, R o , is obtained by summing the contributions from each band : Ro

= lel [{PsPo”(Ptd’”~)a12 ) 2

+ {PsPoS(P!’s/c)al2)31/oo2.

(57)

The addition of low-mass, high-mobility holes greatly influences the weak-field Hall coefficient even though such holes are present in relatively small amounts. Since the mobility weighting factor enters as the square, the Hall coefficient is increased by a factor ofalmost 2 in the 111-V compounds.27 An expression analogous to Eq. (9), which explicitly gives the effect of band structure and scattering on Ro can also be derived from Eqs. (8), (43), (46), and (53). The result for one band is Ro/Rm = poH/p0= (/<7>’)(a,~i2/a:1).

(58)

From this we see that the ratio of zero-field Hall mobility in a warped band to that in a spherical band is PoH/P;,s = a l z l a l l ~

(59)

The infinite-field Hall coefficient, R m ,is simply given by R,

=

l//elc(P,

+ P3).

(60)

At intermediate field strengths, the Hall coefficient is found to be a complicated function of the magnetic field strength in p-type materials that have the Ge-like valence-band structure. This occurs at temperatures where

’’ The actual increase is also dependent on the warping of the bands and on the scattering mechanisms involved. See Beer and Willardson3’ Eq. (22), in the limit of vanishing magnetic field.

18

RICHARD J. STIRN

= lelH.r/m*c 2 1 for one or both carriers, In such a case, a series solution of the Boltzmann equation in powers of H is no longer possible. Instead, McClureZ8has introduced the field dependence by a representation involving Fourier series expansions in harmonics of the frequency of the carrier around the hodograph determined by the intersection of a surface of constant energy in k space with a plane normal to the magnetic field. Closed expressions for the components of the conductivity tensor are impossible except for limiting cases-for example, when the energy surfaces are spheres or polyhedra.28a Goldberg et ~ 1 . ' ~have attempted to account for the field dependence of the Hall coefficient and resistivity in p-Ge by this technique, assuming a spherical light-hole surface, and a cubic heavy-hole surface, but without complete success. Excellent agreement with experiment was obtained by Beer and Willardfor Ge when the warped nature of the energy bands and scattering by acoustical phonons and ionized impurities were taken into account. The final equations are rather cumbersome and will not be presented here. c. Weak-Field Magnetoresistance

The weak-field magnetoconductivity coefficients in Eq. (1) for warped degenerate bands were derived by Mavroides and Lax' with techniques similar to those used to obtain the conductivity and Hall coefficient. From symmetry considerations it can be shown that there may be a maximum of four different nonvanishing components for any cubic crystal.' In practice, however, two of them, o ~ , , ,and , ~ o : ~ ~ always ,,, occur in pairs, so that measurements are determined by at most three independent components. These can be expressed for each band as : oxxXx 0 = -(pSe4/mo3cz)(z3)(A & B')~&?(I-),

4LYy= - ( ~ " e " / m ~ ~ c ~ ) < r ~k>g( )=34a x x y Y , &YX

=

4 Y X Y

=

( p s e 4 / m o 3 c Z ) ( ~ 3? )(A B')3[+uxyxy

+ $@(r)],

(62) (63)

where the anisotropy parameters are given by uxxyy

=

+ o.3838r2 - o.o167r3 + 0.00755r4 + 0.000661r5 - 0.000190r6. .',

I - 0.22141-

J. W. McClure, Phys. Rev. 101, 1642 (1956).

'"R. S.Allgaier, Phys. Rev. 158, 699 (1967); 165, 775 (1968).

C. Goldberg, E. N. Adams, and R. E. Davis, Phys. Rev. 105,865 (1957). A. C. Beer and R. K. Willardson, Phys. Rev. 110, 1286 (1958). 3 1 For additional detail, see Beer'4 pp. 189-212.

z9

30

(64)

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS =

19

I - 0.050or - o.o469rZ + o.oo401-3

+ o.oooii4r5 + o.ooooo4r6 + ..., cqr)= ( i 6 / i m ) r Z ( i - 0.42951- + o.oi88r2 + o.0103r-3 + 0.00249r4 + 0.000474r5 + 0.000085r6 + . ..). -o.ooo63r4

(65)

(66)

It can be assumed that the total magnetoconductivity can be given by coefficients CE,,, which represent the sum of the individual oElmcoefficients for the two kinds of holes present. The weak-field magnetoconductivity is extremely sensitive to the addition of high-mobility holes since the mobility weighting factor enters as the cube. Thus, the magnetoresistance coefficients are strongly field dependent and it is imperative that these be measured in the zero-field limit at all temperatures. Expressions for the directional weak-field magnetoresistance coefficients, Miy, can be written as relations involving the magnetoconductivity coefficients. The results arez5

The term (X$,,Jo0)is the zero-field Hall mobility, p o H = Rooo for both bands, when uo is given by Eq. (51). As was the case for n-type materials earlier, it is convenient to analyze the anisotropy in the magnetoresistance phenomenologically by use of the inverse Seitz coefficients. The magnetoconductivity coefficients can be expressed as functions of b, c, and d by combining Eqs. (67H70)with Eq. (18):

From these relations and Eqs. (55H57), expressions for the inverse Seitz coefficients can now be written. Additional expressions can be obtained which contain only parameters of the energy bands and the energy

20

RICHARD J. STIRN

Similar ratios can be formed using the X;Im coefficient^.^' Note that in the limit of spherical bands (r,+ 0),(b c) and d -+ 0 as expected. Unfortunately, knowledge of these ratios or of the values of b, c, and d does not enable one to determine the band anisotropy as is possible with many-valleyed semiconductors, even with the assumption of only heavy-hole conduction. In principle, one should be able to obtain the warping parameters by determining A and B’ independently, and then choosing a value for C (and thus r)which yields the measured values of the anisotropy ratios in Eqs. (74) and (75). The parameters A and B’ can be obtained from Eq. (45) (neglecting the small correction for anisotropy) and the expressions m* = ( A f B’)-’rn, if any two of the three quantities p 3 / p 2 , m 2 ,and m3 are known. The carrier concentration ratio can be roughly determined by analyzing the magnetic-field dependence of the magnetore~istance.~’*~~ However, uncertainty in the value of ( T ’ ) ’ / ( T ~ > ~ at a given temperature and carrier concentration is too great to make this approach practical. Also, anisotropy in the scattering cannot be separated from the band anisotropy. To the author’s knowledge, theoretical studies of scattering anisotropy are nonexistent for the case of warped energy bands. It has been shown34 that the scattering anisotropy is greater for ionized impurity scattering than it is for intravalley lattice scattering, at least for electrons in silicon. Hence, it may not be unreasonable to assume isotropic relaxation times for the heavy and light holes at temperatures and carrier concentrations where ionized impurity

+

32 33

34

R . J. Stirn and W. M. Becker, Phys. Rev. 148,907 (1966). C. H. Champness, Phys. Rev. Lett. 1,439 (1958). D. Long and J. Myers, Phys. Rev. 120,39(1960).

1.

21

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

+

scattering is expected to be unimportant. In this case, the ratios (b c)/ [b + and [b (p0H)2]/dshould be temperature independent over a range of temperature’ small enough that changes in the band shape should not occur. If theoretical values of the warping parameters A, B, and Cz are available from Kane-band calculation^,^^'^^ the predicted values of the ratios in Eqs. (74) and (75) can be compared with experiment. Conversely, if reliable values of A , B, and C2 are available from cyclotron resonance experiments, information about the relative values of z2 and z3 as a function of temperature can be obtained, with the assumption that the energy dependence of both relaxation times are the same. Up to this point, we have ignored the possible effects of nonparabolicity of the light-hole band. This would introduce a temperature dependence (as the distribution spreads in k space) into the galvanomagnetic properties as the ratio of light to heavy holes rises. The effects would be insignificant at temperatures much below room temperature. Even at higher temperatures, the presence of optical-phonon scattering would minimize any alteration of the predicted properties due to nonparabolicity because the contribution of the higher energy holes to the conductivity is decreased. Any attempt to incorporate this effect into the previous discussion is beyond the scope of this chapter.

+

d. Efective Masses

As we have already noted, holes in the 111-V compounds are normally located at the center of the Brillouin zone in a pair of bands which are degenerate near the point k = 0. Since the constant energy surfaces are warped spheres, the masses are not scalar and the specific value to be used in the simple equations describing the electrical properties depends on the particular property being considered. One example is the density-of-states effective mass md which has been defined by Lax and Mavroides from Eqs. (42) and (43) and given to be24 mdn = [mo/(Af B’)][l -

+ 0.03333rn + 0.01057 L2 - 0.00018rn3

o.oooo3rn4 + . . .I.

Since the total carrier concentration is p

md

md2{

=

p2

+ p3,

-$[mdJmd~I~’~).

(76)

(78)

0. E. Kane, J . Phys. Chem. Solids 1, 82 (1956); in “Semiconductors and Semirnetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 1, Chapter 3. Academic Press, New York, 1966. 36 M. Cardona, J . Phys. Chem. Solids 24, 1543 (1963).

3s

22

RICHARD J. STIRN

Another “type” of effective mass is the conductivity effective mass defined from the relation 0 = pe2(z>/m,. (784 For the warped band, it has been calculated to be24

+

mcn = [m,/(A f B’)][1 o.03333rn - o.oio57rn2 - o.ooo95rn3 +0.00111rn4+

4.

(79)

Referring to Eq. (52),we have 1 if we assume equal relaxation times for both holes. From Eqs. (77) and (SO), we obtain m, = m c 2 [ l ---][Im:i2 mc2 m,3i2 mc3

+m

] ,

md 2

The amount of valence-band warping (and hence the value of r)has been found to be relatively small in Ge,23 Si,23InSb,37and GaSb.38 Theoretical calculations predict a similar situation in the other 111-V corn pound^.^^ Hence differences in the effective masses m, and md are very small. Actually, the linear terms in the energy of the heavy holes are of importance, as well as the effect of higher lying bands on the light-hole mass. Because of these effects, the effective masses have a dependence on temperature and the Fermi-level position. This has led to some confusion about the interpretation of experimental results in p-InSb, for instance. The reader is referred to the reference of Kolodziejczak et for theoretical calculations of the effective masses in 111-V compounds and the influence of the aforementioned effects on them. 111. Experimental Results

3. ALUMINUM ANTIMONIDE a. Band Structure

Aluminum antimonide (AlSb) is a high-energy-gap intermetallic compound semiconductor with a melting point of 1060°C. It crystallizes into a D. M. S. Bagguley, M. L. A. Robinson, and R. A. Stradling, Phys. Lett. 6, 143 (1963). R. A. Stradling, Phys. Lett. 20. 217 (1966). 3 9 J. Kolodziejczak, S. Zukotynski, and H. Stramska, Phys. Status Solidi 14,471 (1966).

37

38

1. BAND STRUCTURE AND GALVANOMACNETIC EFFECTS

k = r r / a (Ill)

k- (000)

23

k = 2*/o (100)

FIG.3. Energy bands of AISb, including spin-orbit splitting, in the [lo01 and [ l 1 1 1 directions. (After Pollak et

cubic zinc-blende structure with a lattice constant of 6.14 A. The preparation and growth of AlSb crystals, which are unstable in the presence of moist air, have been reviewed by Allred.40 Calculations of the band structure for AlSb throughout the reduced zone have recently been made using the pseudopotential method4’ and the k p method.42 Both methods are empirical in nature inasmuch as experimental values of some interband transitions are required. In the k p method, 6 independent matrix elements of an antisymmetrical potential are added to the k - j j Hamiltonian of the isoelectronic group IV element. The antisymmetric potential comes about because of the two dissimilar atoms in the unit cell. The group IV “element” chosen was the hypothetical IV-IV compound Si-cr-Sn, since the lattice constant of AlSb is about midway between that of Si and a-Sn. The results of the k p band calculations for AlSb are shown in Fig. 3. Values for the transitions indicated in the figure have been given by Cardona et u1.,43344and are reproduced in Table 11. These values were obtained from

.

-

-

W. P. Allred, in Willardson and Goering,’ p. 187. M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141,789 (1966). 4 2 F. H. Pollak, C. W. Higginbotham, and M. Cardona, J . Phys. Soe. Japan 21, Suppl. (Proc. Inr. Conf: Phys. Semiconduct., K y o t o , 1966) p. 20. 4 3 M. Cardona, F. H. Pollak, and K. L. Shaklee, Phys. Rev. Lett. 16, 644 (1966). 44 M. Cardona, K. L. Shaklee, and F. H. Pollak, Phys. Rev. 154, 696 (1967). 40

41

24

RICHARD J . STIRN

TABLE I1 ENERGYSPLITTINCS FOR AlSb (300°K) Transition EO

E, &I’

E2

Energy (eV) 2.22 2.86 3.72 4.25

1

Transition A0

A1 43‘

6

Energy (eV) 0.75 0.40 0.27 0.36

electroreflectance measurement^,^^'^^ which give more resolution than ordinary reflectivity The value of the minimum energy gap E , was first determined from optical absorption data by Blunt et who reported a value for E, of (1.633.5 x 10-4T)eV. These authors also obtained a O”K value of 1.60eV from a linear extrapolation of the intrinsic resistivity (above 750°C) as a function of temperature. Turner and Reese4* obtained a similar temperature coefficient from their optical absorption measurements, but their value for E , was 0.1 eV higher. Blunt et aL4’ detected an absorption band at 0.75 eV (1.6 p) in p-type material, which was observed and correctly interpreted by B r a ~ n s t e i nas~ ~due to transitions between the split-off valence band and the uppermost valence band (Ao). An absorption band present at about 4 p (0.3 eV) in n-type material47 was thought by Blunt et al. to be caused by a deep donor level, while Turner and R e e ~ attributed e~~ it to transitions from the lowest-lying conductionband minima to a higher band a t k = 0. Paul5’ ruled out indirect transitions as the explanation for the 4 p band on the basis of pressure measurements on an analogous band occurring at 3 ,u in Gap. It is now b e l i e ~ e d ~that ~ , ~this ’ infrared peak is due to X I - X , transitions (6). The presence of higher-lying conduction minima can be detected under some circumstances by measuring the photoresponse of surface barrier contacts on the semiconductor. Mead and Spitzer5’ resolve such data into 45

A. Frova and P. Handler, Phys. Rev. 137, A1857 (1965); Phys. Rev. Lett. 14, 178 (1965).

B. 0. Seraphin, R. B. Hess, and N. Bottka, J . A p p l . Phys. 36, 2242 (1965). 46’See, e.g., B. 0.Seraphin, in “Semiconductors and Semimetals,” (R. K. Willardson and A. C. Beer, eds.), Vol. 9. Academic Press, New York, 1972. 4 7 R. F. Blunt, H. P. R. Frederikse, J. H. Becker, and W. R. Hosler, Phys. Rev. 96, 578 (1954). 48 W. J. Turner and W. E. Reese, Phys. Rev. 117,1003 (1960). 4 9 R. Braunstein, Bull Arner. Phys. SOC.4, 133 (1959). 5 0 W. Paul, J . A p p l . Phys. 32,2082 (1961). 5 1 R. Zallen and W. Paul, Phys. Rev. 134, A1628 (1965). 5 2 C. A. Mead and W. G. Spitzer, Phys. Rev. Lett 11, 358 (1963). 46

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

25

three different valence-to-conduction-band transitions with threshold energies of 1.50, 1.85, and 2.10eV. On the basis of these results and a reevaluation of (Al,Ga, -,)Sb alloy data,52aMead and Spitzer identify the band minima, in order of ascending energy, as [loo], [ill], and k = 0. The absolute values of the energies above the valence-band maxima should not be taken strictly since the interpretation of their data involves the assumption of an unproven model for the photoinjection from the metal contact.

b. Galvanomagnetic Eflects n-Type When grown without any intentional doping, AlSb is always p-type, with an acceptor concentration of about 1OI6 cm-3 for the purest material grown to date. Selenium, or more commonly Te, is added to the melt in order to obtain n-type material. The Hall coefficientand resistivity as a function of temperature are given in Fig. 4 for Te-doped AISb. The donor activation energy eD, extrapolated to O’K, is 0.068 & 0.001 eV.53*54 The Hall mobility versus temperature for the same sample used for Fig. 4 and for a more highly compensated sample is shown in Fig. 5. The calculated temperature dependence of the mobility, assuming combined polar optical, acoustical mode, and ionized impurity scattering, was found55 to be in reasonable agreement with the measured dependence. However, there is a growing realization that intervalley scattering is the dominant mechanism limiting electron lattice mobilities in the indirect gap compounds, while polar mode optical phonon scattering remains dominant in direct gap corn pound^.^^" Consequently, the role of acoustical mode scattering is reduced, and thus also, the need for deformation potentials higher than what seem reasonable. Besides the photoresponse measurements by Mead and Spitzer,” there is additional indirect evidence that the conduction-band minima in AlSb are not at the zone center. Edwards and D r i ~ k a r n e rfound ~ ~ a red shift of the absorption edge under pressure with a pressure coefficient of - 1.6 x eV kg- ’ cmP2.From an analogy with silicon, the authors concluded that the optical transitions near the absorption edge weredue to [ 1001conduction-band 52aI. I. Burdiyan, Fiz. Tuerd. Tela 1, 1360 (1959) [English transl.: Sou. Phys.-Solid State 1, 1246 ( 1960)]. 53 F. J. Reid, in Willardson and Goering,’ p. 158. 5 4 R. J. Stirn and W. M. Becker, Phys. Rev. 141,621 (1966). 5 5 R. J. Stirn and W. M. Becker, J . Appl. Phys. 37, 3616 (1966). 55aD.L. Rode, private communication; see also reference 97a. 5 h A. L. Edwards and H. G. Drickamer, Phys. Rev. 122, 1149 (1961).

26

RICHARD J. STIRN

t

I u

m\

5

f

U

I2

w

I!

LL LL

w

0 V .A

1

a

I

4

IOOO/T,

O K - ’

FIG.4. Hall coefficient, resistivity, and n T 3 I 2 versus reciprocal temperature for AISb. (After Stirn and B e ~ k e r . ’ ~ )

minima. Piller and Patton5’ found no change in sign of the interband Faraday rotation throughout the range of wavelengths used. This and the fact that the Faraday rotation shifts to shorter wavelengths with decreasing temperature, as in silicon, indicated that the transitions were indirect. The first direct evidence of the [loo]-type conduction band minima in AlSb was obtained from piezoresistance studies.58 It was found that thr values of the elastoresistance coeficients for n-A1Sb were similar to those 57 58

H. Piller and V. A. Patton, Phys. Rev. 129, 1169 (1963). K. M. Ghanekar and R. J. Sladek, Bull. Amer. Phys. SOC.10, 304 (1965).

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS loo0

I

I

I

I I I l l

\

I

I

I

27

I

m 0

2

n - Al Sb (Te) 0 8 41 I n300.K 0 D 21 I nmeK

20

40

= 5 X l0l6 ~ = 2 x lot7

r n - ~

100 200 TEMPERATURE, "K

400

FIG.5. Hall mobility versus temperature for AlSb of two different carrier concentrations. (After Stirn and B e ~ k e r . ~ ~ )

values found for n-Si and that the symmetry relations expected in the deformation-potential theory of electron transfer between [loo] ellipsoids were approximately satisfied. However, when Ghanekar and Sladek calculated the anisotropy parameter K from the piezoresistance and piezo-Hall effect data, they obtained a value of 18,59 which is near the value for Ge. One would expect a value of K closer to 5, considering the analogy of AlSb with Si. Magnetoresistance measurements have now been reporteds4 that give more reasonable values of the valley anisotropy. The angular variation of the magnetoresistance for some representative crystal orientations is given in Fig. 6. The magnetoresistance is very small because of the low electron mobility, but the data points have little scatter. The horizontal curve shows the transverse magnetoresistance for a sample with current flow along the cubic axes. Inspection of Eq. (17) indicates that M& (transverse) is equal to 59

K. M. Ghanekar and R. J. Sladek, Phys. Rev. 146,505 (1966).

28

RICHARD J . STlRN I

I

I

I

n - A I S b (Te) H.25 Kg

0.0030

I

I 0 X

I

I

I

1

1

D211 HI. <110> T 195 "K D21I H I < l i O > 1 1 ; <110>

0.0025

0.0020 90

\

4

a

0.0015

L.-r.

A D 311 T=245"K

0.0010 -

I /I

H1<001>

H / I

0.0005

I

0"

I

I

I

20"

40"

60"

I

I 100"

80"

I 120"

I 140"

I 160"

I 180'

[HI FIG.6 . Angular variation of magnetoresistance in n-A1Sb for two samples at different temperatures. The upper curve for sample D211 represents transverse to longitudinal changes in orientation. (After Stirn and B ~ c k e r . ~ ~ )

the inverted Seitz coefficient b, independent of magnetic-field direction. Thus the data ofsample D311 in Fig. 6 show the absence ofspurious magnetoresistance effects due to contact ~ h o r t i n g ' * ~and ' ~ inhomogeneities.20,20a The magnetoresistance was reported to be proportional to the square of the magnetic field up to at least 30 kG. Values of the inverted Seitz coefficients b, c, and d were reported by Stirn and B e ~ k e r . 'The ~ symmetry relations b + c = - d , d < 0 [Eq. (23)] for [loo]-type spheroids were obtained for samples from two ingots of different carrier concentration and for temperatures ranging from 77 to 295°K. Application of Eq. (29) and choice of the prolate spheroidal model (since A > 1) gave values of the anisotropy parameter K + at the different bath temperatures (Fig. 7). The decrease in K + with decreasing temperature is expected since the ratio T , , / T ~is > 1 for ionized impurity scattering34 in many-valley semiconductors. Similar behavior has been observed in n-Ge.60 This explanation is supported by the fact that K + is consistently lower for sample D211, which has greater ionized impurity scattering as 6o

B.A. Laff and H. Y. Fan, Phys. Rev. 112,317 (1958).

1.

/*::”:::

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

’-

10

i’: 6-

1

I

1

I

1

I

1

-

xD2II

-

-

4-

I

-

3-

2

-

n-AISb (Te)

06411

I

Y

1

29

I

I

I

I

I

I

I

1

determined from Hall effect and mobility data.54 The extrapolated value of K + for lattice scattering is 7 f 1, which is comparable to the value of 5 for Si. The electron effective mass in AlSb is commonly quoted as 0.39m0. This value was determined by Moss et aL61 from Faraday rotation of free carriers. If we assume that zl,/zI = 1 in AlSb at temperatures above 300°K where K + = 7, application of Eq. (37) yields effective-mass component values of mll =

1.64m0,

m,

=

0.23m0.

(82)

The conductivity effective mass m,* [Eq. (35)] is then 0 . 3 3 and ~ ~ the ~ densityof-states effective mass md* [Eq. (36a)l is l.5mo, if one assumes six valleys as in silicon. The value of m,* agrees well with that obtained from infrared reflectivity measurement^^^ (0.30mo). The density-of-states effective mass can be obtained from thermoelectric-power measurements if the scattering mechanism is specified. Nasledov and Slobodchikov62 determined an average value of md* equal to (1.2 0 . 4 ) by ~ ~assuming acoustical-mode scattering in their Se-doped AlSb samples. Stirn and Becker’ have made calculations of the electron mobility temperature dependence in AlSb, based 61

62

T. S. MOSS,A. K. Walton, and B. Ellis, in Proc. Int. Conf Phys. Semiconduct. Exeter p. 295. Inst. Phys. Phys. SOC.,London, 1962. D. N. Nasledov and S. B. Slobodchikov, Fiz. n e r d . Tela 1, 748 (1959) [English transl.: Sou. Phys.-Solid State 1, 681 (1959)l.

30

RICHARD J. STIRN

on a combination of ionized-impurity7 acoustical-mode, and polar-opticalmode scattering, that suggested a value for md* of l S m , if no additional scattering mechanisms are present. It should finally be noted that the calculated effective-mass values above assumea value ofmll/m, = 7, i.e., T ~ , / T * = 1. Ifthe latter equality is somewhat different because of the presence of lattice scattering anisotropy above 300"K, all values quoted would be reduced by a small amount. Pollak, et aL4' calculated mass parameters at a number of points in the Brillouin zone with eigenvectors of the Hamiltonian used to obtain Fig. 3. These parameters are given in Table 111. There are no experimental values for the mass parameters in the higher-lying minima. Their calculated value of m, in the [loo]-minima agrees well with that reported by Stirn and B e ~ k e r , 'but ~ the m l l * value of 0.95mO7which is substantially lower, makes their mass anisotropy parameter in the lowest minima equal to 3.8. The actual value is probably somewhere between 3.8 and 7. Because of the uncertainties in the experimental values of m,* and mF* and their relative insensitivity to K for low values of K , cyclotron-resonance measurements will be required to resolve the difference. c. Galvanomagnetic EfSects p-Type

The lowest room-temperature carrier concentrations in p-A1Sb grown to date are about 10l6 cmP3.Hall coefficient and resistivity data32as a function

TABLE 111 ELECTRON EFFECTIVE-MASS PARAMETERS FOR AlSb IN FREEELECTRON MASS

U N I T S OF THE

Pollak et ~ 1 . 4 ~ Pollak ef Pollak et al. 42 Pollak et Stirn and B e ~ k e r ~ ~ Pollak et ~ 1 . ~ ~ Stirn and B e ~ k e r ~ ~ Moss el ~ 1 . ~ ' Eq. (35) with K = 7, m , * ( X , , A , ) = 0.23 Eq. (36) with K

six valleys

=

7, mI*(Xl, A , )

= 0.23,

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

31

T. "K 4x1013 4x104 p-AlSb

H.25

kG

400

200 2

6

10

14

22

103/~, OK-[

FIG. 8. Hall coefficient, resistivity, and p T 3 I 2 versus reciprocal temperature for p-AISb. (After Stirn and B e ~ k e r . ~ ' )

of temperature are shown in Fig. 8 for a temperature range of 50-500°K. The activation energy of the dominant acceptor is 0.033 eV. The acceptor impurity is not known for sure, but carbon and copper are possibilities. It is likely that the acceptor levels in AlSb are nearer to the band edge than those in silicon by a factor of 1.5-2.63 On the basis of a model with two carriers having different mobilities, one would normally expect the Hall coefficient to be strongly dependent upon the magnetic-field strength. Strong field dependence is indeed seen in p-A1Sb 63

W. P. Allred, W. L. Mefferd, and R. K. Willardson, J . Electrochem. SOC.107, 117 (1960).

32

RICHARD J. STIRN

MAGNETIC FIELD, kG

FIG.9. Hall coefficient factor R J R I S k Gversus magnetic field strength for p-AISb at T = 50, 77, and 113°K. (After Stirn and B e ~ k e r . ~ * )

as shown in Fig. 9. The monotonic decrease in Hall coefficient with increasing magnetic field is also observed in p-Ge and very pure P - S ~ . ~ ' , ~ ' The theory developed for this behavior in Ge and Si,30331 which was described briefly in an earlier section, has been applied to P - A ~ SThe ~ . results ~~ are shown in Fig. 10. A measure of the amount of ionized impurity scattering is given by pi = 6 ~ ~ , ~ / The p , , ~values . of the carrier concentration ratio p3/pz [Eq. (45)] and the lattice mobility ratio pL,3/pL,2 [Eq. (53)] were calculated to be 0.085 and 5, respectively, using warping parameter values64 of A

=

5.96,

B

=

3.36,

C 2 = 23.2.

(83)

More recent calculation^^^ give a value for C 2 which is reduced by nearly a factor of 2 from that given above. However, the latest value would not predict the observed field dependence of the Hall coefficient because we would obtain p 3 / p 2 = 13% and pL,3/pL,2= 3.7. The dashed curve in Fig. 10 shows a calculation for pL,3/pL,2= 3, the value for Si. A strong maximum is seen to occur with increasing field strength. Initially, R , increases as H increases because of the warping of the heavy-hole band ;at higher fields, the light hole 64

Private communication to R. J. Stirn and W. M. Becker by M. Cardona.

1.

BAND STRUCTURE AND GALVANOMACNETIC EFFECTS

33

MAGNETIC FIELD, kG

FIG.10. Comparison of theoretical Hall coefficient factor R,/R, with experimental data for p-A1Sb at a temperature of 77°K. (After Stirn and B e ~ k e r . ~ ' )

enters the strongfield region and produces a decrease in R,. Except for ultrapure material, this latter behavior has also been observed in p-type Si.30 Since the shape of the field dependence of R , is so sensitive to the value of pL,3/pL,2, and to a lesser extent, to the value of p 3 / p 2 , the larger value of C2 seems to be more a p p r ~ p r i a t e . ~ ~ " The Hall mobility temperature dependence for p-type AlSb is shown in Fig. 11. The temperature dependence is given by a T - ' . 9 5 law above 200°K. Because of the magnetic-field dependence of R,, very high magnetic fields are required in order to accurately measure the temperature dependence of R , and p, at lower temperatures. Fields in excess of 100 kG seem to be necessary for p-A1Sb with room-temperature carrier concentrations of about 10'6cm-3 (see Fig. 10). A recent calculation64bsupports this contention. The results of the k .p calculation, which used pseudopotential formalism to calculate the required matrix elements, gave a value of 20.9 for C2. 64b R. L. Bowers and G. D. Mahan, Phys. Rev. 185, 1073 (1969).

64a

34

RICHARD J . STIRN 10

I

I

I 1 1 1 1 1

I

I

I

I .

p - AlSb

p2950K=3.~ x 1016cm-3

H = 2 5 kG

TEMPERATURE "K

FIG.11. Hall mobility versus temperature for p-A1Sb. (After Stirn and Be~ker.~')

The lattice mobility ofholes in 111-V compounds was recently examined by Wiley and D i D ~ m e n i c o . ~They ~ " conclude that the polar mode mobility expressions derived for nondegenerate s-like bands are not applicable to the degenerate p-like valence bands, and that the temperature dependence of the hole mobility can be better explained by combining acoustic and nonpolar optical mode scattering alone as has been shown in Ge and Si. Their calculations, which included AlSb, use the acoustical deformation potential as an adjustable parameter. The magnetoresistance coefficients in p-A1Sb show a strong magneticfield dependence at lower temperatures as shown in Fig. 12. The angular variation of the magnetoresistance is shown in Fig. 13 for the current along 64cJ. D. Wiley and M. DiDomenico, Phys. Reu. B2,427 (1970).

1.

35

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

MAGNETIC FIELD, kG

FIG.12. Transverse magnetoresistance divided by H Zversus magnetic-field strength for p-AlSb at T = 50,77,and 113°K. (After Stirn and Be~ker.~') I

I

/

\ 0"

20"

40"

60"

80"

100"

120"

140"

160"

180"

CHI

FIG.13. Angular variation of magnetoresistance in transverse and longitudinal position for p-AlSb at T = 77°K. (After Stirn and Becker.")

36

RICHARD J. STIRN

a [110] direction. A ratio of transverse to longitudinal magnetoresistance ' (extrapolated to zero magnetic field) of about 7 has been r e p ~ r t e d . ~The existence of a nonzero longitudinal magnetoresistance is consistent with warping of the heavy-hole band. TABLE IV EXPERIMENTAL VALUESOF SOMEWEAK-FIELD MAGNETORESISTANCE COEFFICIENTS FOR p-AISb"

~~~

50°K

77°K

113°K

195°K

245°K

295°K

348 338 46 348 -292 -20

238 216 31 238 - 185 -44

135 118 21 135 -97.0 -34

34.0 29.0 6.3 34.0 -22.7 -10

19.4 16.3 3.6 19.4 -12.7 -6.2

12.1 10.3 2.0 12.1 -8.4 -3.5

~

MY::

(cm" V-' ~;:O,(cm"v-' M , , , (cm" V-' b (cm4 v-' c(cm" V-' d (cm" V-'

x lo-" sec-Z) x lo-" sec-') x sec-') x lo-" sec-') x lo-" sec-*) x

After Stirn and B e~ k e r . ~ '

The weak-field inverted Seitz coefficients (Table IV) were found to obey the relations ( b + c ) z -d,

d <0,

(84)

in the higher temperature range. The magnitudes of (b + c) and d diverge more the lower the temperature. Frequently, the equality ( b c) = - d , (d < 0) is reported in the literature for p-type materials that have valence bands similar in nature to those of germanium. For a multivalley-band model, this equality indicates [loo] valleys. However, piezoresistance measurements on such p-type material give results indicating [l 111 valleys. Since the same results are observed in p-type Ge, this seemingly contradictory observation is interpreted to mean that the material under investigation has a Ge-like valence band. However, an inspection of Eqs. (74) and (75) shows why the symmetry relation (b + c) = - d is approximately correct for a model of warped valence bands. In the limit of negligible g n(actually B,, is about two orders of magnitude smaller than the other anisotropy parameters aijkr[see Eqs. (64H66)]), it is seen that the ratio (b + c ) / ( - d ) approaches unity.32 The anisotropy factor B,, is proportional to the square of r whereas the other factors aijklare near unity. It would appear that changes in r reflect the discrepancy in Eq. (84) at lower temperatures. Changes in r would signify changes in the band parameters with temperature. However, such changes would not be significant in the temperature range reported. The

+

1.

37

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS TEMPERATURE, "K

0 l4

p-AlSb

INGOT # 5 2

FIG. 14. Weak-field magnetoconductivity coefficients for p-AISb versus pO2a,. (After Stirn and Be~ker.~')

theory of Mavroides and Laxz5 assumes an isotropic relaxation time, and hence, assumes that r is only dependent on the band warping parameters (e.g., the effective mass). Since magnetoresistance measurements do not separate the anisotropic effects of mass and relaxation time, it is more likely that the above observed behavior at lower temperatures reflects large changes in the relaxation-time anisotropy, rather than changes in the band parameters. This argument was strengthened by further manipulation of the expressions for the magnetoconductivity coefficient^.^^ The absolute values of the relaxation times can be eliminated if it is assumed that the energy dependence of both relaxation times is the same.65 The coefficients Cijlmare then found , the to be directly proportional to A(poZao),where A = ( T ) ( T ~ ) / ( T ~ ) ~and proportionality constants contain the anisotropy factors (aijkl)and Bn. Figure 14 shows the dependence of the magnetoconductivity coefficients where the latter quantity was varied by changing the temperature. on (pLo2a,), All of the data points lie on a straight line, except those for XxxXxat the three 65

P. J. Kernmey and E. W. J. Mitchell, Proc. Roy. SOC.(London)A263,420(1961).

38

RICHARD J. STIRN

lowest temperatures. Since changes in A with temperature should be the same for all three coefficients, and since only C ,,,, is proportional to Bn and thus to r2 [see Eqs. (61H63)], the above argument concerning anisotropy in the relaxation times appears pertinent. Use of Eq. (79) and the warping parameters in Eq. (83) gives values of the conductivity effective mass for the heavy hole (0.48m0) and the light hole (0.10m0). The density-of-states effective masses [Eq. (76)] are (0.52m0) and (0.10m0), respectively. The amount of warping of the heavy-hole band is small enough that there is little difference between the two types of effective masses. The light- and heavy-hole masses can also be estimated directly36 from the E , and E,' gaps given in Table 11. Cardona et obtained m2* = 0.52m0 and m3* = 0.11m0. Experimental determinations of the hole masses are scarce and in disagreement. The effective mass of heavy holes in AlSb has been estimated by Reid and Willardson66 from the variation of mobility with carrier concentration. With the assumption of acoustical-mode scattering for the lattice mechanism, they obtained a value of 0.4m0. However, higher values have been reported on the basis of thermoelectricpower measurements. Sasaki et d6'obtained a room-temperature value of 1.8 f 0.8 for (m2*/m,)g213,where g is the degeneracy of the band edge. With g = 4, we get m2* = (0.7 0.3)m0. Nagledov and S l o b o d ~ h i k o v ~ ~ report a value of m2* = (0.9 f O.l)m, from thermoelectric-power measurements between 400 and 700°K. Both groups assumed the lattice scattering mechanism to be acoustical mode.

4. GALLIUM PHOSPHIDE a. Band Structure

Gallium phosphide (Gap), with its considerably higher melting-point temperature of 1470°C, is more difficult to prepare in the crystalline form than AlSb. Some methods used in the growth of Gap, which crystallizes into a cubic zinc-blende structure with a lattice constant of 5.45 A, have been reviewed by Miller.69 Many of the crystals used in the investigations reported below were grown by vapor-transport technique^.^^".^^^ 6b

F. J. Reid and R. K. Willardson, J . Electron. Control 5, 54 (1958).

'' W. Sasaki, N . Sakamoto, and M. Kuno, J . Phys. SOC. Japan 9,650 (1954).

D. N. Nasledov and S. V. Slobodchikov, Zh. E k h . Fiz. 28, 715 (1958) [English transl.: Sou. Phys.-Tech. Phys. 3, 669 (1958)l. 6 9 J. F. Miller, in Willardson and Goering,' p. 194. "T.J . Frosch, J . Electrochem. SOC. 111, 180 (1964). 69bA.S. Epstein and W. 0.Groves, Advan. Energy Conversion 5, 161 (1965). b8

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

39

Calculations of the band structure for GaP have been performed with the same techniques as for AlSb. The group IV “element” used as a basis for the crystal potential was the hypothetical compound Ge-Si. The results of the E jj band calculations for G a p are shown in Fig. 15. The electroreflectance spectrum of G a p at room temperature is shown in Fig. 16. Values for the transitions indicated in Fig. 15 are given in Table V. References to these and additional values are given below. TABLE V

ENERGYSPLITTINGS FOR GaP (300°K) Transition

E,’ E2 a

Energy (eV)

4.71 - 5.3”

Transition

Energy (eV)

--

0.082

0.08 0.06 0.28“

See text.

Values for the energy of the indirect gap were first reported by Folberth and O~wald,~’ who obtained a room-temperature value of 2.24 eV from optical absorption data. Similar values have since been r e p ~ r t e d . ~ l * ~ l - ~ ~ Recently, the optical absorption edge has been measured in exceptionally good single crystals for temperatures between 1.6 and 300”K.74Absorption components associated with four different phonon energies were resolved, enabling the authors to derive precise values for the indirect gap of 2.259 f 0.003 eV at 300”K75and 2.339 k 0.002 eVat 1.6”K.A similar type ofmeasurement has been made more recently by Lorenz et at temperatures up to 900°K. The authors obtained a value of 2.261 eV at 300°K. The indirect-gap temperature coefficient AE$AT has been commonly 0. G. Folberth and F. Oswald, 2. Naturjorsch. 9a, 1050 (1954). M. Gershenzon, D. G. Thomas, and R. E. Dietz, Proc. Int. Con$ Phys. Semiconduct. Exeter, p. 752. Inst. Phys. Phys. SOC.,London, 1962. 7 2 W. K. Subashiev and S. A. Abagyan, in “Physics of Semiconductors” (Proc. 7th Int. Conf.) p. 225. Dunod, Paris, 1964. 73 D. N. Nasledov, V. V. Negreskul, S. I. Radautsan, and S. V. Slobodchikov, Sou. Phys.-Sotid State 7,2965 (1 966). ’4 P . J. Dean and D. G. Thomas, Phys. Reu. 150,690 (1966). 7 5 This value has since been raised by approximately 3 meV on the basis of measurements of 70

71

the temperature shift of the absorption band due to indirect excitons weakly bound at nitrogen impurities (Dean et 75aM. R. Lorenz, G. D. Pettit, and R.C. Taylor, Bull. Amer. Phys. SOC.13,453 (1968);Phys. Rev. 171,876 (1968).

RICHARD J . STIRN

FIG.15. Energy bands of Gap, including spin-orbit splitting, in the [loo] and [ l l l ] directions. (After Pollak et d4’)

PHOTON ENERGY, e V

FIG. 16. Electroreflectance spectrum of n-type GaP at room temperature. (After Thompson eta[.

19)

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

41

stated in the literature to be -5.2 x 10-4eVoK-'.51976However, recent measurements7 (see reference 75) have indicated a substantially lower room-temperature coefficient of -(2.36 & 0.01) x eV OK-'. This dependence is linear only down to about 140°K. A general form of the energy gap temperature dependence [E,(O) - aT'/(T b)] has been proposed by Var~hni~ on~ the " basis of electron-phonon interactions rather than lattice dilation being the cause for the temperature shift. Empirical values of a = 6.2 x eV OK-' and p = 460°K for G a p have been derived very recently77bfrom absorption measurements taken from room temperature up to 1273°K.The functional relationship also fits the lower temperature data of Lorenz et ~ 1 . quite ~ ' ~well down to 6°K. The very weak reflectivity peak associated with the direct transition E,(Tr -,r,c)was first detected by Zallen and Paul," who reported a value for E , given by 2.78-4.6 x lOP4(T- 300")eV at temperatures above 80°K. Nelson et aL78 obtained similar results from photoconductivity-response measurements, deducing a value of 2.895 2 0.002 at O"K. A quadratic form for the temperature dependence of -(1.17 k 0.01) x 10-6T2 eV OK-' was reported, however, Neither the linear nor quadratic form appears to describe the temperature variation of the direct gap for the entire range of 0-300°K. Later s t ~ d i e s ~ ' , have ~ ~ , ~given ' values of the direct gap comparable to those above. Recently, however, extremely thin crystals of GaP ( 51 p) have been grown, enabling Dean et al. to make very precise measurements of the intrinsic optical a b ~ o r p t i o n They . ~ ~ deduced a value for the direct-exciton energy gap at 25°K of 2.8725 _+ 0.0005 eV. At 295°K the energy gap has a value of 2.780eV and has a temperature coefficient of -(5.2 & 0.1) x eV OK-'. This temperature coefficient and the value of the indirectgap coefficient reported by Dean et al.,77 which is more than one half less than previously reported values, means that the energy of the [100]-minima in GaP shifts upwards with decreasing temperature less rapidly than that of the zone center rl minimum, contrary to previous belief. A more energetic absorption band at 2.954 eV was also detected by Dean et al.77 at low temperatures. Assuming that this is due to direct transitions from the split-off component of the valence band, the spin-orbit splitting A, is 0.082 eV. Previous estimates of this splitting, 0.10 eV72*79 and 0.127 eV,80,8'

+

F. Oswald, Z . Naturforsch. 10a, 927 (1955). J. Dean, G. Kaminsky, and R. B. Zetterstrom, J . Appl. Phys. 38, 3551 (1967). 77aY. P. Varshni, Physica 34, 149 (1967). 77bM. B. Panish and H. C. Casey, Jr., J . Appl. Phys. 40, 163 (1969). '13 D. F. Nelson, L. F. Johnson, and M. Gershenzon, Phys. Rev. 135, A1399 (1964). '9 A. G. Thompson, M. Cardona, K. L. Shaklee, and J. C. Woolley, Phys. Rev. 146,601 (1966). 8 o J. W. Hodby, Proc. Phys. Soc. (London) 82,324 (1963). 8 1 M. L. Belle, Zh. I. Alferov, V. S. Grigor'eva, L. V. Kardinova, and V. D. Prochukhan, Fiz. Tverd. Tela 8, 2623 (1966) [English transl.: Sou. Phys.-Solid State 8,2098 (1967)l.

76

" P.

42

RICHARD J . STIRN

are higher. Since the splitting is not expected to be appreciably temperature dependent, the reason for the discrepancy is unknown at this time. There has been considerable controversy about the source of the strong absorption peak and reflectance structure at 3.7 eV. The peak, previously" assigned to E,' (r;5 ri5), has been reinterpreted as El (A3 --t A,) on the basis of GaAs-GaP alloy data,72*79-81-8pseudopotential band calculat i o n ~ ,and ~ ~ photoelectric-emission studies.84 The linear temperature coefficientof El between 80" and 300°K is about - 3.5 x eV oK-1.81,8s A discussion of the El peak and the associated spin-orbit splitting A1 is given by Thompson et ~ l . who , ~ report ~ the only experimental value for Al( -0.08 eV), in agreement with the calculated value of 0.072 eV obtained from the k p method (Fig. 15). l-i 5 ) Various investigator^^^,^'-^^ have given values for the E,' (l-7 transition ranging from 4.75 to 4.8 eV. This transition has a linear temperature coefficient of about -3.4 x 10-4eV0K-1 between 80 and 295"K.82*8s Thompson et point out that the main contribution to E,' and E,' + A,' probably comes from a region of k space in the [loo] direction near k = 0. However, since the corresponding conduction and valence bands are nearly parallel in this region, the energy at which E,' occurs should well represent the value of the r;5 -+ r;, gap. The spin-orbit splitting A,' associated with the E,' peak is approximately 0.06 eV.79 6(X5' X3') are Values for the transitions E,(X,' + Xlc) and E , very difficult to deduce accurately from the various optical measurements in the literature. In fact, assigning any peak in the reflectance spectrum to the E , or E , 6 transition is hazardous in any of the III-V compounds, judging from the results found in GaAs.86 The problem would seem to come from interband transitions associated with a much larger region of the reduced zone, which dominate in their contribution to the main reflectivity peak (or photoemissive dip). In addition, the determination of absorption thresholds of transitions to X,' from the zone center are imprecise, partly because of phonon interactions and low density-of-states. Interpretations of the well-known infrared absorption band, which is related to 6 and discussed below, have also been confused because of the effects of free carrier absorption and photoionization of occupied donor levels. A peak in the reflectance (and electroreflectance) spectrum of GaP at about 5.3 eV has been ideptified with the E , transition at room temperature.44~79~82~83*8s~87 A similar -+

-

-+

+

-+

+

J. C. Woolley, A. G. Thompson, and M. Rubenstein, Phys. Rev. Lett. 15,670(1965). T. K. Bergstresser, M. L. Cohen, and E. W. Williams, Phys. Rev. Lett. 15, 662 (1965). 8 4 T. E. Fisher, Phys. Rev. 147,603 (1966). 8 5 A. G. Thompson, J. C. Woolley, and M. Rubenstein, Can. J . Phys. 44,2927 (1966). 8 6 F. Herman and W. E. Spicer, Phys. Rev. 174,906 (1968). H. Ehrenreich, H. R. Philipp, and J. C. Phillips, Phys. Rev. Lett. 8, 59 (1962). 82

83

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

43

identification has been made from photoelectric-emission m e a s ~ r e m e n t s . ~ ~ of The transition has a linear temperature -4.5 x 10-4eVoK-’. derived a value of 4.92 eV for the E, gap (Fig. 15) from their Pollak et k p calculations, and suggest that the 5.3 eV peak is due to the saddlepoint transitions & + C, (1 10 axes), which was calculated to be 5.40 eV. They further suggest that the Eo’ and E , transitions are almost degenerate at about 4.8-4.9 eV. The most recent theoretical using an empirically adjusted OPW band calculation, assigns to the E , gap a value of 4.6 eV. However, band calculations have not demonstrated the accuracy necessary for a meaningful comparison with the optical results above. The above calculations of Pollak et al.42also do not account for the presence of a 5.74 eV peak in the electroreflectance s p e c t r ~ m (5.66 ~ ~eV~ dip ~ ~in the photoemissive response84) at room temperature. These maxima were identified with the X,’ + X,” transition, thus giving a value of 0.46 eV44379 and 0.38eV84 to 6, the conduction band splitting at X . However, these values would be expected to be larger than the true X I c.+ X 3 “gap because of the problem of broadened response due to multiple transitions. That this is so seems to be confirmed by measurements of the infrared absorption band due to the direct XIc+ X,” transition. The peak of the infrared band in this region occurs at 3.0 p (0.40 eV).87b,88This energy has often been quoted erroneously for the value of 6, perhaps influenced by an early published theoretical value of 0.40 eV.41 More accurate, of course, is the extrapolated threshold of this band. First estimates of this value gave 0.31 eV87band 0.30 eV.” However, the total infrared absorption includes absorption due to free carriers, which must be properly subtracted. This in turn involves a knowledge of the dominant scattering mechanisms at the temperature of measurement. The result of subtracting a linear extrapolation of the free carrier absorption (FCA) at longer wavelengths, where the FCA is proportional to wavelength, was 0.33 eV.88aMost recently a nonlinear subtraction of the theoretical FCA utilizing acoustic and nonpolar optical deformation-potential scattering was performed on the experimental absorption and a value for 6 of 0.276 0.007 eV was obtained.89

-

87eF.Herman, R. L. Kortum, C. D. Kuglin, and J. P. Van Dyke, “Methods in Computational Physics” (B. Adler, S. Fernback, and M. Rotenberg, eds.), Vol. 8, p. 193. Academic Press, New York, 1968. 87bW. G. Spitzer, M. Gershenzon, C. J. Frosch, and D. F. Gibbs, J . Phys. Chem. Solids 11, 339 ( 1959). J. W. Allen and J. W. Hodby, Proc. Phys. SOC. (London) 82, 315 (1963). ssaYu. V. Shmartsev and A. D. Remenyuk, Fiz. Tekh. Poluprou. 3, 1697 (1969) [English transl.: Sou. Phys.-Semicond. 3, 1425 (1970)l. 8 9 J. D. Wiley and M. DiDomenico, Phys. Rev. B1,1655 (1970).

44

RICHARD .I. STIRN

The values for 6 quoted above were all obtained from room-temperature absorption data. The absorption threshold data at lower temperatures, where most of the free carriers are frozen out into the donor levels, do not require the subtraction procedure. However, one would expect to get photoionization of the dominant donors, and hence, obtain a somewhat higher value for 6 from the absorption data. This has indeed been observed with the additional energy being about 40-60 meV.87b,88,90Thus the X,' + XIctransition appears to be nearly temperature independent as might be expected. One further confirmation of the lower value of 6 is the direct measurement of the optical absorption due to transitions from the zone center r to Xlcand X,' individually (at low temperature) with the resulting difference in energy of 0.29 eV.77 Two small, but distinct, peaks with energies of 6.67 and 6.90eV were observed in the reflectance of These were identified as El' and El' AI' peaks attributable to L,' + L,' transitions (not shown in Fig. 15).Unlike the splitting at k = 0 (Ao), the spin-orbit splitting of the L point was found to vary little with x.This fact suggests that the splitting of L is due mainly to the gallium cation.

+

b. Galvanomagnetic EfSects : n-Type Most electrical measurements on GaP are performed on crystals grown epitaxially by open-tube vapor transport. These can be grown n-type by intentionally doping with Se, Te, or S. The choice of the substrate is quite crucial according to a very recent investigation by Taylor et aLgl Samples grown on 1 1 1A faces of either GaP or GaAs crystals are always high-resistivity p-type even when doped with S or Se, whereas samples grown on 111B faces are always n-type, including those intentionally undoped. The residual impurity in the undoped crystals was shown to be sulfur at a concentration of 2-3 x 10'6cm-3. In addition, it was found that samples grown on a GaAs substrate (which is the usual choice because of the scarcity of GaP crystals of adequate size) are less uniform in carrier concentration and have somewhat lower Hall mobilities. The observation about the effect of the 11 1A face on the type of conductivity would explain the findings of Kamath and Bowman92 who always obtained p-type conduction in their undoped GaP crystals. Hall-effect measurements were performed on crystals prepared and doped by the floating-zone process for one of the earlier systematic determinations 90

91 92

A. D. Remenyuk, L. G. Zabelina, Yu. I. Ukhanov, andYu. V. Shmartsev, Fiz. Tekh. Poluprou. 2,666 (1968) [English transl.: Sou. Phys.-Semicond. 2, 561 (196811. R. C. Taylor, J. F. Woods, and M. R. Lorenz, J . Appl. Phys. 39, 5404 (1969). G. S. Kamath and D. Bowman, J . Electrochem. Sor. 114, 192 (1957).

1. BAND

STRUCTURE AND GALVANOMAGNETIC EFFECTS

45

of donor activation energies in The samples had varying degrees of compensation and showed considerable scatter in their extracted activation energies. The average values of activation energies (for a donor concentration of 1OI8 ~ r n - were ~ ) taken to be 0.089,0.078, and 0.076 eV (f0.020 eV) for S, Si, and Te, respectively. The values are expected to be larger for lower donor concentrations. Montgomery94 has reported Hall measurements on Te-doped GaP with improved homogeneity. The results gave a binding energy of 80 _+ 5 meV for Te at a concentration N , 3 x lo" ~ r n - ~ . When the binding energy was plotted versus N;l3, an extrapolation to zero concentration resulted in a value of 95 meV. Other measurement^^^ on Tedoped crystals with higher mobilities (170 cm2 V-' sec-' at room temperature) led to an activation energy of 0.11 eV for Te as deduced from Hall measurements. Greatly improved values of donor (and acceptor) ionization energies have been deduced from the recombination luminescence spectra of excitons bound to neutral donors in S, 0.102 eV; Te, 0.0875 eV; Se, 0.100eV; Si (donor), 0.080eV, and more recently, Sn, 0.0655 eV.95c The technique gives very precise values for the donor-acceptor pair energy separation, which is not sensitive to the relatively high impurity concentrations in Gap. However, the absolute values for the ionization energies do depend upon proper interpretation of the spectra, in particular, a correct assignment to the transition involved. As a result of observation of infrared absorption due to excitation of electrons bound to S and Si donors95dand Te donorsg5=in Gap, correct values for the ionization energies of these impurities were directly obtained. As a consequence, it was determined that the excited states in the twoelectron spectrum are s-like rather than p-like. The effect of stress on the two-electron spectrum95e was found to be consistent with the new interpretation, which was first suggested by F a ~ l k n e rThe . ~ ~revised ~ values are : S, 104.1 & 0.3meV; Te, 89.8 0.3meV; Se, 102.8 2 0.6meV; and Si (donor), 82.1 0.3 meV. The above value for Sn already reflects the new interpretation.

-

H. C. Montgomery and W. L. Feldmann. J . Appl. Phys. 36, 3228 (1965). H. C. Montgomery, J . Appl. Phys. 39,2002 (1968). 95 D. N. Nasledov, V. V. Negreskul, and S. V. Slobodchikov, Fiz. Tuerd. Tela 7 , 1912 (1965) [English transl.: Sou. Phys.-Solid State 7, 1549 (1965)l. 95aP. J. Dean, J. D. Cuthbert, D. G. Thomas, and R. T. Lynch, Phys. Rev. Lett. 18, 122 (1967). 95bP. J. Dean, C. J. Frosch, and C. H. Henry, J . Appl. Phys. 39, 5631 (1969). '"P. J. Dean, R. A. Faulkner, and S. Kimura, Phys. Rev. B2,4062 (1970). 95dA.Onton, Phys. Reu. 186,786 (1969). 95eA.Onton and R. C. Taylor, Phys. Rev. B1,2587 (1970). "'R. A. Faulkner, Phys. Rev. 184,713 (1969). 93 '4

46

RICHARD J. STIRN

104

103

t

E

0 I

c Q -.

102

10

0

I

2

3

4

5

6

7

I OOO/T

8 O K

9

-'.+

10 I I

12 1314

I

FIG.17a. Hall effect and resistivity as a function of temperature for n-type Gap. (After Ep~tein.'~)

The temperature dependence of the Hall constant and resistivity for n-type GaP is shown in Fig. 17a for an epitaxially grown crystal with a total impurity concentration of 3.1 x 1OI6cm3. Room-temperature values of the Hall mobility of about 150 cm2 V-' sec-' and a temperature dependence T-" with x = 1.9 at the higher temperatures are typical of the purest crystals grown to date, although values up to 180-190 cm2 V - ' sec-' for the roomtemperature mobility have been reported The temperature dependence of the electron mobility in several samples of Te-doped GaP has been analyzed by Toyama et ul.97afor various scattering mechanisms (Fig. 17b). The authors found that intervally scattering with LA phonons near the X-point with a characteristic temperature of 300°K

'' A. S. Epstein, J . Phys. Chem. Solids 27, 1611 (1966). T. Miyauchi, H. Sonomura, and N. Yamamoto, Jap. J . Appl. Phys. 6, 1409 (1967) 97aM.Toyama, M. Naito, and A. Kasami, Jap. J . Appl. Phys. 8,358 (1968).

97

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

47

10

10

7"

8

5 cu

E i

1 0'

0 OBS

- CALC

50

70

100

200

300

500

700

T, " K

FIG.17b. Analysis of electron mobility versus temperature for GaP with various scattering mechanisms. (After Toyama et OL''~)

was the principle mechanism limiting the mobility from 200 to 400°K. They were able to fit the experimental data within that temperature range by combining intervalley scattering with acoustical-mode, piezoelectric, and polar-mode scattering. Continued good fit with experiment down to a temperature of 77°K was achieved by including space-charge region (hard

48

RICHARD J . STIRN

sphere) ~ c a t t e r i n g Other . ~ ~ authors have also incorporated this latter type of scattering in Gap.95,96,99 As pointed out in the section on AlSb, the role of polar-mode scattering of electrons is apparently of little importance in indirect gap III-V compounds, in agreement with the work of T ~ y a m a . ~ ~ ~ Ionized impurity scattering appears to play only a minor role in limiting the electron mobility at lower temperatures. It has been reported97399that impurity conduction sets in at temperatures around 60"K, depending upon the impurity concentration. The usual phenomena associated with impurity conduction, e.g., a maximum in the Hall constant, a change in slope of the resistivity curve, and the appearance of a negative magnetoresistance, were observed in both undoped and doped GaP near the temperature mentioned above. In addition, a transition from phonon-assisted hopping process to metallic conduction occurred at a critical concentration of 2 x 1OI8 cmP3. However, the samples used in these investigations were not removed from their GaAs substrates and such behavior was not observed by Taylor et aL9' in their crystals grown on GaP substrates even at doping levels greater than 10l8 ~ m - ~ . No magnetoresistance measurements for n-type GaP have been reported up to this time. As in AlSb, Ap/po will be very small because of the large amount of impurity scattering present in the crystals that are available now. The only experimental evidence concerning the location of the conduction minima in k space is indirect, albeit convincing. Measurements on GaAs-GaP alloys reveal a nonlinear variation of energy gap with composition."' This fact and the functional form of the dependence of absorption coefficient upon photon energy above threshold8' indicate that the transition is indirect. This conclusion is supported by the large electron effective mass, which is discussed below. That the minima lie along the [loo] directions in k-space was suggested by the pressure dependence of the band gap, which shifts to lower energies, as in s i l i ~ o n . ~ ~ ~ ' ~ ' The electron effective mass in GaP has been determined to be mF* = (0.35 k 0.02)m0 from measurements o f the Faraday rotation.61.'02 The conductivity effective mass has recently been obtained from infrared reflectivity measurements on heavily doped Gap (Te)lo3: m,* = (0.32 t- 0.02)m0 with n = 8.9 x lo'* cm-3 and mc* = (0.36 & 0.03)m0 with n = 1.4 x 1019 y8

L. R. Weisberg, J . Appl. Phys. 33, 1817 (1962).

" T.

Hara and 1. Akasaki, J. Appl. Phys. 39, 285 (1968). 0.G. Folberth, 2. Naturforsch. 10a, 502 (1955). 101 A. L. Edwards, T. E. Slyhouse, and H. G. Drickamer, J. Phys. Chem. Solids 11, 140 (1959). lo' Yu. I. Ukhanov and Yu. V. Mal'tsev, Fiz. Tverd. Tela 5, 1548 (1963) [English transl.: Sov. Phys.-Solid State 5, 1124 (1963)l. lo' U. Zhumakulov, Fir. Tverd Teln 8, 3099 (1966) [English transl.: Sou. Phys.-Solid State 8, 2476 (1967)J 100

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

49

cm-3, and from reflectivity measurements on heavily doped GaP (S)lo4: m,* = 0.35m0 with n = 8.5 x lo1*~ m - Unfortunately, ~ . no experimental value for the anisotropy parameter K is available. If we assume a value of K = 5 as in silicon, application of Eq. (37) yields effective-mass component values of m l l *= 1.12m0, m,* = 0.22m0. The calculated value of m,* [Eq. (35)] is then 0.31m0, in good agreement with the experimental value. The density-of-states effective mass md* [Eq. (36a)l is 1.25mo, if we assume six valleys in the conduction band in the [loo] direction occurring at a value of k somewhat less than that at the zone edge (see Fig. 15). Pollak et ~ 1 calculated . ~ ~mass parameters at a number of points in the Brillouin zone for Gap. Their results and the values just discussed are given in Table VI. Note that if K = 7, as it may be for AISb, the second value for mil*(XI, Al) is in agreement with the calculated value.42 [The component m,*(X,, A l ) is little affected, decreasing by O.O1mo.] In this case, we get m,* = 0.29m0 and md* = 1.33m0. A fitting of Faulkner's effective-mass calculations for donor statesgsdto the observed levels in the infrared excitation spectra of GaP yielded electron mass parameters of m l l *= (1.7 L- 0.2)m0 and m,* = (0.191 f 0.005)m0 for Si and S donors,gsband m l l *= (1.5 k 0.2)m0 and m,* = (0.180 & 0.005)m0 for Te donors.95c However, both sets of parameters yield low values of mF* (0.31m0) and m,* (0.26m0). TABLE VI ELECTRON EFFECTIVE-MASS PARAMETERS FOR GaP FREEELECTRON MASS

IN

UNITSOF

THE

Pollak et Pollak et Pollak et Pollak et Eq. (37) with K

=

5 and mF* below

0.28 0.22

Pollak et a/.42 Eq. (37) with K

=

5 and mF* below

mF*

0.35

Moss et

m,*

0.31

Eq. (35) with K = 5, m,*(X,, A l ) = 0.22

*

1.25

Eq. (36) with K = 5, m , * ( X , , A j ) = 0.22

m,*(X,, A , )

md

Io4

six valleys

M. Hashimoto and 1. Akasaki, Phys. Lett. 25A, 38 (1967).

50

RICHARD J. STIRN

Taylor et aL9' combined their Hall data with that of Montgomery and Feldmann' to derive a mean value for the density-of-states effective mass per valley of 0.42m0. With the assumption of six valleys, the total mass md*is 1.39m0,in good agreement with the calculations above for 5 < K < 7. c. Galvanomagnetic Efects : p-Type

Gallium phosphide which exhibits hole conduction is often made by doping crystals with Zn. Additional dopants which act as acceptors are Mg, Cd, C, and Ge at concentrations below 10I8 cm-3.105Copper diffusion has been used to obtain high-resistivity GaP'06*'07 crystals which were used for photoconductivity measurements. The presence of a "sensitizing" hole trap at about 0.7 eV was attributed to the copper acceptor. Hall measurements have given conflicting activation energies for the various acceptors; probably due to the presence of varying amounts of compensation, neglect of the T 3 / 2factor, in some cases, and neglect of the magnetic-field dependence of the Hall coefficient which is due to light-hole conduction and heavy-hole band warping. As an example, the activation energy for Zn has been reported to be 0.031,97 0.040,'08.1090.051,1'0 and 0.060 0.002 eV,"' while the luminescence study by Dean et aZ.,95awhich was referred to earlier in regard to donor levels, resulted in a value for of 0.062 5 0.002 eV.' l 2 Also reported95a were the acceptor binding energies of 0.095 eV for Cd and 0.046 eV for Si. Silicon is known to be an amphoteric impurity in 111-V compounds. Thus, in Gap, Si will be a donor when substituted on a Ga site and will be an acceptor when substituted on a P site. A luminescence study by Lorenz and Pilkuhn'14 first suggested that Si was a donor in GaP with a level between the Te and Se donor levels, and also, that F. A. Trumbore, H. G. White, M. Kowalchik, C. L. Luke, and D. L. Nash, J. Electrochem. SOC.112, 1208 (1965). Io6 B. Goldstein and S. S. Perlman, Phys. Rev. 148, 715 (1966). lo' D. Bowman, J. Appl. Phys. 38,568 (1967). l o * H. G. Grimmeiss and H. Koelmans, Phys. Rev. 123, 1939 (1961). l o g M. Gershenzon and R. M. Mikulyak, Solid State Electron. 5, 313 (1962). M. M. Cohen and F. D. Bedard, J. Appl. Phys. 39,75 (1968). ' ' I H. C. Casey, Jr., F. Ermanis, and K. B. Wolfstirn, J. Appl. Phys. 40,2945 (1969). "* Very recent Hall-effect studies on zinc-doped GaP'1',113, seen to indicate an excess hole concentration as compared to the Zn concentration. The ratio of free holes to added Zn was as much as 1.5 throughout the concentration range (5 x 1OI6 to 1 0 ' 9 ~ m - 3 when ) hole concentrations were calculated from p = l/R,e. The effect, important for an understanding of one of the predominate luminescent processes (Cd or Zn-0) in Gap, may be caused by additional acceptor defects which are introduced simultaneously with the Zn, or more likely, is caused by valence band warping, making the Hall factor r < 1. ' I 3 L. M. Foster, J. F. Woods, and J. E. Lewis, Appl. Phys. Lett. 14, 25 (1969). 'I4 M. R. Lorenz and M.H. Pilkuhn, J. Appl. Phys. 38,61 (1967). '05

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

51

16

''1

2

3

4

5

6

7

IOOO/T,

8

9

I O I I

12

OK-'

FIG.18. Carrier concentration ( x ) and resistivity (0) as a function of temperature for p-type Gap. (After Cherry and Allen.'I6)

Si was a deep acceptor of about 0.25 eV when substituted on a P site. These observations were affirmed recently by Dean et al.95bwho show that the 0.046 eV level' " is actually due to carbon and that there is a deep acceptor level at 0.204 (kO.002)eV due to Si. Recently pair spectra involving the acceptors Be and Mg have been observed in GaP and binding energies of 0.0535 eV for Mg' 5a and 0.050eV for Be' 5b were reported. The temperature dependence of the carrier concentration and resistivity in p-type G a P is given in Fig. 18. The highest room-temperature hole mobilities reported to date have been 14092and 150116cm2 V - ' sec-'. The temperature dependence of the mobility has been calculated''1 with the

'

The acceptor ionization energies quoted in the text for Dean et dgSa have been revised95h upward by 2 meV because of the use of an updated value for the dielectric constant. Thus, E,(C) = 0.048 eV, E,(Zn) = 0.064 eV, and E,(Cd) = 0.097 eV. J. Dean, E. C. Schonherr, and R. B. Zetterstrom, J . Appl. Phys. 41,3475 (1970). llsbP.J. Dean and M. Ilegems, Bull. Amer. Phys. SOC.15, 1342 (1970). l 6 R. J. Cherry and J. W. Allen, J . Phys. Chem. Solids 23, 163 (1962). 'lS

52

RICHARD J. STIRN

assumption of polar mode optical phonon scattering and ionized and neutral impurity scattering. Only fair agreement with the mobility ofa sample having a room-temperature mobility of 120 cm2 V-' sec- was found (Fig. 19a) for rn*/rn, = 1.0. However, as discussed earlier for p-AlSb, it is now believed that polar scattering is not significant in p-type 111-V compounds. A combination of acoustic and nonpolar optical mode (NPO) scattering has been shown to give a much better temperature dependence fit as well as a more realistic mobility value.64' The difference in temperature dependence, particularly for T 2 0.60, where 0 is the optical phonon characteristic temperature (580°K for Gap), between the polar mobility and the mobility resulting from combined acoustic and NPO mode scattering is clearly seen

D'

2

4

6

lo2

2

4

6

lo3

TEMPERATURE T. OK

FIG. 19a. Calculatedtemperature dependence(dashedline) of the hole mobility in Gap. (After a s e y eta[."')

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

0.I

1.0 T/B

53

10

FIG.19b. Comparison between the temperature dependence of the polar mobility and the mobility resulting from combined acoustic and nonpolar optical mode scattering. (After Wiley and D i D ~ m e n i c o . ~ ~ " )

in Fig. 19b. Bowman'07 derived an electron-to-hole mobility ratio of about 1.2 from correlation of photoconductance and photo-Hall data in p-type Gap, in good agreement with experimental values. Limited magnetoresistance measurements for p-type GaP have been reported.'I7 At T = 128"K, the magnetoresistance was proportional to the square of the magnetic field for fields between 1 and 8 kG. In analogy with p-type AlSb (Fig. 12), deviation from this dependence because of two-carrier conduction would be expected at higher fields. The authors also observed a large increase in A p / p o as the temperature decreased and attempted to based on pure polar-mode compare the magnitude with predictions' scattering. However, the incorporation of impurity scattering and light-hole conduction is crucial to any quantitative comparison. In any case, the contribution of polar scattering to the lattice mobility of holes is thought to be negligible.64" Experimental details of the valence-band structure are not available up to this time. We can estimate some features by using warping parameters which have been derived from k p calculation^.^^ We will take for these

'

-

'" D. N. Nasledov and S. V. Slobodchikov, Fiz. Tuerd. Tela 4, 2755 (1962) [English transl.: Sou. 'I8

Phys.-Solid State 4,2021 (1963)l. B. F. Lewis and E. H. Sondheimer,Proc. Roy. SOC. (London) 227A, 241 (1955).

54

RICHARD J . STIRN

parameted4 those values obtained from the calculations used to derive the corresponding values for AlSb [Eq. (83)] : A

=

4.5,

B

=

2.1,

and

C 2 = 20.1.

(85)

More recent calculation^^^ give a value for C 2 which is reduced by nearly a factor of three from that given here. The parameters in Eq. (85) indicate somewhat more warping of the heavyhole band than in AlSb. If one neglects the small differences between the density-of-states and conductivity masses, use of the above parameters gives a heavy-hole effective mass of 0.58~1,and a light-hole effective mass of 0.14m,. The ratio of light-to-heavy holes [Eq. (45)] is 0.094 and the corresponding lattice mobility ratio pL,3/pL,2 is 4.2 [Eq. (53)]. All of these values are not too unlike those of AISb. There have been no experimental determinations of the effective masses ; however, values of the (heavy) hole mass of O S 1 l 6 and 0.6rn,' l o have been used in hole-mobility studies, and a density-of-states mass of approximately 0.5 rn, has been estimated from the temperature dependent hole concentration.' l 1

5. ALUMINUM ARSENIDE The knowledge of the properties of aluminum arsenide (A&) and of the compounds still to be discussed is scarce indeed, even when compared to AlSb and Gap, due to extreme difficulties in preparing high-purity material. These difficulties are : (1) very high melting-point temperatures, (2) high chemical reactivity with the melt containers, and (3) high decomposition pressure at the melting point. AlAs has a melting-point temperature of 1700°K. It crystallizes into small crystallites with a cubic zinc-blende structure of lattice constant 5.66 A. The preparation of AlAs has been reviewed by Stambaugh."' AlAs and the other A1 III-V compounds react with moist air. Because of the general systematic trend of the relative energies of the three conduction-band minima with average atomic number, extrapolation from silicon suggests that the lowest minima in AlAs are [lo01valleys.50No experimental evidence about the nature of the band structure is presently available. The band gap obtained from earlier optical absorption data at room temperature is 2.16 eV.120,121The photovoltaic response of surface barrier contacts on AlAs as reported by Mead and Spitzer" is shown in Fig. 20. The solid circles give the square root of the photovoltaic response as a function of photon energy. At energies below the band gap, the photoresponse is due to photoinjected electrons from the metal contact. By extrapolating this 119

E. P. Stambaugh, in Willardson and Goering,' p. 184. P. H. Keck, private communication to F. Herman (1955). G . A. Wolff, R. A. Hebert, and J. D. Broder, Phys. Rea. 100, 1144 (1955).

1.

.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

f

2

-

2ol---151

x

P

-

~

w

-

z

-

In 0

1.4

1.6

1.8 2.0 2.2 2.4 2.6 hu, eV

55

2.8 3.0

FIG.20. Photovoltaic response of surface barrier contacts on AIAs. The straight lines are the result of the subtraction procedure described in the text. (After Mead and Spit~er.~')

response to energies greater than the band gap and subtracting from the total response, the data shown by the open circles are obtained, representing valence-to-conduction-band-minimatransitions. The 2.1-eV value for the indirect transition is in good agreement with the optical results given above. A higher transition at 2.9 eV, identified by the authors as the direct one, is obtained by subtracting off the indirect response and the metal-contact response. More recent optical absorption data have been reported for AlAs over a wide temperature range.121a*121b The absorption edge was found to be dominated by indirect optical transitions consistent with the conduction band minima at or near X. The value of the indirect band gap was measured to be 2.16 eV at 300% 2.223 eV at 77"K, and 2.238 eV at For the first time p n junctions have been formed in AlAs by diffusing Zn into n-type vapor-grown AlAs layers.121cThe devices were found to emit visible light peaking at 2.146 eV at room temperature with forward biasing. An independent estimate of the band gap was made by extrapolating the p n photovoltage as a function of the incident light energy to zero voltage. 121sM. R. Lorenz, R. Chicotka, G. D. Pettit, and P. J. Dean, Solid-State Commun. 8,693 (1970).

"lbW. M. Yim, 1. Appl. Phys. 42,2854 (1971). 121cC. J. Nuese, A. G. Sigai, M. Ettenberg, J. J. Gannon, and S. L. Gilbert, Appl. Phys. Lett. 17, 90 (1970).

56

RICHARD J . STIRN

Energy gaps determined in this manner were 2.15 eV at room temperature and 2.23 eV at 77°K. Herman et al. made a nonrelativistic OPW band calculation using the Kohn-Sham exchange approximation. 1 2 2 The computations were performed at the zone points r, X , and Land used the experimental values of the indirect and direct gap to empirically refine the first-principles approach. Their results for AlAs gave splittings of 4.8 eV for E,' (I-; -, r; 5 ) , 4.2 eV for E , (X,' -+ XIc),4.9 eV for E , + 6(X,' -+ X3'), 3.6 eV for El' (L3' + Llc), and 6.0eV for El' + A,'(L3' L3'), using the empirical values of 2.2 and 3.0 eV for the indirect and direct gap, respectively. In these values, the spinorbit splittings are represented by their weighted means. Stukel and Euwema'22ahave also performed a first principles nonrelativistic OPW band calculation for AIAs. However, their calculation used Slater's free-electron-exchange approximation and was done in a self-consistent manner using only the lattice parameter as an adjustable parameter. The values for the splittings agree with those given above'22 within a few tenths of an electron volt. In addition, the computed values for the room temperature indirect and direct gaps, are 2.26 and 2.40 eV, respectively. Evidence of the indirect nature of the minimum energy gap has recently been reported on the basis of optical a b ~ o r p t i o n and ' ~ ~ piezore~istance'~~" studies on the Al,Ga, -,As system. The piezoresistance measurements indicated electron conduction in (100) minima for x > 0.4, and hence, by inference in AlAs. In the absorption study an abrupt change of slope between the two linear regions of the gap versus composition curve occurred at 2.0 eV and 50 mole % AlAs, and the extrapolated value for the direct transition in AlAs was 2.6 eV. However, the authors concluded that the gap in this system does not follow a linear interpolation between the direct transition minima. While no detailed calculation of the valence-band structure has been attempted, it can be assumed that the valence band is not unlike that of the other 111-V compounds. By using a simple model which employs the free atom spin-orbit splittings of the constituent atoms of AlAs, Braunstein and Kane2 have estimated the spin-orbit splitting to be 0.29eV and the light-hole effective mass to be 0.22m0 in AlAs. Very little has been reported on the electrical properties of AlAs. Carrier concentrations on the order of 1019 cmP3 and resistivities of l o p 2 to --f

F. Herman, R. L. Kortrum, I. B. Ortenburger, and J. P. Van Dyke, Final Rep., Contract No. F33615-67-C-1793.Project No. 7885, Wright-Patterson Air Force Base, Ohio. 1z2aD. J. Stukel and R. N. Euwema, Phys. Rev. 188, 1193 (1969). 12' J. F. Black and S. M. Ku, J . Electrochem. SOC. 113,249 (1966). 1 2 3 a J . C. McGroddy, M. R. Lorenz, and J. E. Smith, Jr., J . Appl. Phys. 42, 1852 (1971). lZ2

1.

57

‘ooor--T

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

62

I0

100

IN

TEMPERATURE, O K

FIG.21. Temperature dependence of the Hall mobility of n-type AIAs. (After Whitaker.”’)

R-cm have been measured’24 in rather impure material, which exhibited p-type conductivity. More recent measurements, made between 59 and 400°K on singlecrystal epitaxially grown layers, indicated n-type conductivity’ s and showed no magnetic-field dependence upon the Hall voltage. However, there was evidence of impurity band conduction at temperatures below 100°K. An analysis of the carrier concentration at temperatures higher than this value was made’” using the relation n(n + NA)/(ND- N , - n) = (2/g)(2nm*kT/h2)3/2e-ED’kT. (86) The activation energy obtained from the Hall data was 0.018 eV and the level associated with this value was tentatively attributed to Si. The best fit of Eq. (86) to the data was made when the value ND - N , = 1.7 x 1OI8 cmW3and m* = 0.5mo ( g = 2 ) were used. The rough estimate of the electron effective mass is consistent with conduction in [1001-type minima. Because of the high doping, the value of 0.018 eV obtained for the Si donor activation energy should be low. Recent photo luminescence measurements on AI,Ga, -,As (0.8 5 x 5 0.95)12’“indicate that ionization energies of the shallow donors Te and Sn are about 0.060 eV. The data also gave an ionization energy of 0.056 k 0.005 eV for the acceptor Zn. The temperature dependence of the electron mobility1” is given in Fig. 21. The highest mobility (180 cm’ V-’ sec-’) was measured on the sample used in the analysis above.

‘

V.N. VertoprakhovandA.G.Grigor’eva,Izu.Vysshikh. Llcheb.Zaued. Fiz.1958(5),133(1958). J . Whitaker, Solid State Electron. 8, 649 (1965). ‘”‘H. Kressel, F. H. Nicoll, F. Z.Hawrylo, and H. F. Lockwood, J. Appl. Phys. 41,4692 (1970). lZ4

lZ5

58

RICHARD J. STIRN

6. ALUMINUM PHOSPHIDE

Aluminum phosphide crystallizes into the zinc-blende lattice with a lattice parameter of 5.45 A.It has a melting-point temperature of more than 2000°C. Methods of reacting the components at these high temperatures and at the high phosphorus vapor pressures required have been reviewed by Rabenau.lZ6 Since AIP is isoelectronic with Si, whose band structure is well known, several authors have attempted to relate the band structure of AIP to that of Si. The first such attempt was by Herman, l Z 7 who developed a semiempirical perturbation scheme neglecting the spin-orbit interaction. He found that the conduction-band minima lie along the (100) axes as in Si. A similar conclusion was reached by Bassani and Yoshimine,’” who applied the orthogonalized plane-wave (OPW)method”* and obtained a value of4.7 eV for the band gap. Three recent OPW band calculations place the value of the forbidden band width near 2.4 eV, a value which, as we will see later, has also been obtained experimentally. The first of these three c a l c u l a t i ~ n s ’ ~ ~ used an expansion of 70 plane waves obtained by perturbing the silicon potentials, and resulted in a band-gap value of 2.4 eV. The other two calculations are first-principles OPW-band calculations. In one,’30 no adjustable parameters are used, although the lattice constant is needed. Stukel and Euwema obtained good convergence using 229 plane waves, finding that the Slater exchange model yielded more accurate results when compared to experiment than did the Kohn and Sham exchange model. Some energy eigenvalues are shown in Table VII, where it is seen that the indirect gap [loo] was found to be 2.1 eV and the direct gap 3.3 eV. The other firstprinciples band calculation,’z2 not done in a self-consistent manner, found that the Kohn and Sham exchange approximation gave more accurate results than did the Slater approximation, contrary to the conclusions of Stukel and Euwema.13’ The computed energy levels are shown in Table VII where it is seen that the two sets of calculations agree within 0.1 to 0.4eV. ~ ~ ’ also empirically refined the first principle computations Herman et ~ 1 . have by assuming an exact value for the indirect gap (r:5 + X I c )of 2.7 eV, but the resulting energy-level values show less agreement with the values in the first column. These empirically modified values are not presented here since A. Rabenau, in Willardson and Goering,’ p. 181. F. Herman, J. Electronics 1, 103 (1955). See also F. Bassani, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 1, p. 64. Academic Press, New York, 1966. 1 2 9 A. S. Poplavnoi, Fiz. Tuerd. Tela 8, 2238 (1966) [English transl. : Sou. Phyx-Solid State 8, 1179 (1967)l. I 3 O D. J. Stukel and R. N. Euwema, Phys. Rev. 186,754 (1969).

lZ6

I*’

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

59

TABLE VII ENERGY-LEVEL STRUCTURE OF Alp BASEDON OPW-BANDCALCULATIONS Level

X3E

X I c(indirect gap)

X,' X3'

XI'

Slater exchange"

Kohn-Sham exchangeb

4.8 3.3 0.0 -11.5

5.2 3.1 0.0 - 11.4

2.9 2.1 -2.1 - 5.3 - 9.2

3.1 2.3 - 2.0 - 5.0 - 9.2

5.0 4.2

5.1 4.3

8.8 5.3 3.0 - 0.8 - 5.5 - 9.8 6.1 3.8

8.9 5.6 3.4 - 0.7 - 5.2 - 9.8 6.3 4.I

x3c - X 5 " ( E 2+ 6)

xic- X,'(E,) LIE L3c

LIC L3"

Ll' Ll'

L3' L,'

-

L,'(El' L,'(E;)

+ Al')

"From Stukel and E ~ w e m a . ' ~ ~ From Herman rt a[.'22

there now seems to be good experimental data pointing to a forbidden band width of 2.4 eV at room temperature. These data, which have been reported by Grimmeiss et ~ l . , ' ~ 'were obtained from optical reflectance measurements and gave a band-gap value of (2.54-4 x 10-4T) eV. The room-temperature gap of 2.42 eV also corresponds to a peak in the spectral curve of the photoconductivity which they measured in the same crystals. In addition, the authors found electroluminescence in bands centered at 5550 A (2.25 eV) and 6150 A (2.03 eV) and calculated activation energies of 0.15 and 0.37 eV from the conductivity temperature dependence (Fig. 22). If we assume that the latter levels (near the valence band) are responsible for the electroluminescence bands, and neglect the Franck-Condon principle, a band gap of 2.40 eV follows from the data. H. G. Grimmeiss, W. Kischio, and A. Rabenau, J . Phys. Chem. Solids 16, 302 (1960)

60

RICHARD J. STIRN

01

I

2

I

3

I

4

I

I

5

103

6

I

7

I

SJ

OK-'

FIG.22. Conductivity versus temperature in undoped p-type AIP. (After Grimmeiss et

Earlier optical absorption data gave a value of 3.0 eV for the band gap, ~ 'should ~ ~ ~ ' ~be ~ perhaps due to the use of highly impure ~ r y s t a l s . ' ~ ~ * ' ~ ~It noted that the value of 2.4 eV, rather than 3.0 eV, most closely follows the empirical rule that the band gap of the zinc-blende crystal is twice that of its corresponding diamond-type analog. The lower value of 2.4 eV has been corroborated very recently by two independent measurements of optical absorption. "la,' 32a Also luminescence studies of InAlP alloys have given an estimate of 3.6 eV for the direct gap of As in other 111-V compounds, the valence-band structure is assumed to be similar to that of Ge and Si. Braunstein and Kane2 have estimated the spin-orbit splitting A, to be 0.05 eV, whereas Stukel and Euwema13' obtained 0.023 eV (compared to the value of 0.044 eV for Si). The light-hole effective mass has been calculated to be 0.392m0 .2 Small crystals of undoped AlP were grown by Grimmeiss et al.13' as both n-type and p-type. Their activation energies for the dominant acceptor levels have already been given (Fig. 22), while the value obtained in n-type AlP was reported to be (0.02eV. No Hall measurements were made. Other crystals of AIP were grown epitaxially on Si and GaAs substrates with vapor transport technique^.'^^ The samples were always n-type and carrier concentrations from 5 x 10l8to 5 x l O I 9 were obtained from Hall measurements. The highest Hall mobility was 60 cm2 V-' sec- at room temperature. The mobility decreased with decreasing temperatures down to 77°K. while the Hall coefficient remained constant. H. J. Hrostowski, Bell. Lab. Rec. 34,246 (1956). 132aB.Monemar, Solid-state Commun. 8, 1295 (1970). 132bA.Onton and R. J. Chicotka, J. Appl. Phys. 41,4205 (1970). 133 F. J. Reid, S. E. Miller, and H. L. Goering, J. Electrochem. SOC.113,467 (1966). 132

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

61

7. ALUMINUM NITRIDE Aluminum nitride (AIN), the last of the four aluminum compounds discussed here, is the most difficult of the four to grow in a crystalline form suitThe principal measurement able for fundamental investigations.' 34,1 problems are absorbed oxygen and the very small size of the crystallites. A1N is different from the other compounds discussed so far in that its crystal structure in hexagonal wurtzite (a = 3.1 11 A, c = 4.978 A), and also, in that it appears to be quite stable in moist air and does not oxidize at temperatures below 700°C. AIN does not form a melt ;rather, it sublimes at a temperature somewhere near 2400°C. Thus, the preparation of AlN crystals involves some type of vapor process or sintering. Two types of crystals are found to be formed, depending upon the processing temperature, platelets and six-sided prismatic needle~.'~'These are always colorless or some shade of blue. The coloration of the blue crystals is caused by the presence of aluminum oxycarbide (A1,OC) which is amorphous with A1N.'35,'36 Epitaxial growth of AIN on substrates of hexagonal silicon carbide with thicknesses up to 25 p has also been reported.' Optical absorption data indicate an indirect band gap ; however, because of additional absorption, apparently due to oxygen, the value of the gap energy is not well known. The centers of these oxygen absorption bands were found to lie at 4.53 and 4.80 (f0.08) eV, and the indirect gap was estimated to be equal to or greater than 3.5eV in one i n ~ e s t i g a t i 0 n . l ~ ~ Lagrenaudie' 39 reported E, = 3.8 eV based on absorption measurements. Andreeva et ~ 1 . ' ~ determined ' a gap width of 4.26 eV from the temperature dependence of the resistivity between 1100 and 1470°K. These data can be compared with a semiempirical estimate of 4.3 eV for the band gap by Wang et al.,14' who compared AlN to the corresponding group IV-IV material (p-Sic). The absorption edge due to the direct transition is much more discernible and has been measured to be about 5.8eV at room t e m p e r a t ~ r e . ' ~ ~ , ' ~ ' A. Rabenau, in Willardson and Goering,' p. 174. K. M. Taylor and Camille Lenie, J . Electrochem. SOC.107, 308 (1960). 13' G. Long and L. M. Foster, J . Amer. Ceram. SOC.42, 53 (1959). 13' T. L. Chu, D. W. Ing, and A. J. Noreika, Solid-state Electron. 10, 1023 (1967). IJ8 J. Pastrfiak and L. Roskovcova, Phys. Status Solidi 26, 591 (1968). 13' J. Lagrenaudie, J . Chim. Phys. 53, 222 (1956). 140 T. V. Andreeva, I. G. Barantseva, E. M. Dudnik, V. L. Yupko, Teplof: Vysok. Temp. 2, 829 134 135

14*

(1964). C. C. Wang, M. Cardona, and A. G. Fischer, R C A Rev. 25, 159 (1964). G. A. Cox, D. 0. Cummins, K. Kawabe, and R. H. Tredgold, J . Phys. Chem. Solids 28,543 (1 967).

62

RICHARD J. STIRN

of Pastrfiak and Roskovcova'38 also estimate a spin-orbit splitting about 0.14 eV from their polarized light absorption spectrum. Experimental results regarding more energetic transitions and theoretical calculations of band structure have not been published for AIN. There is no reason to believe, however, that the band structure is any different qualitatively from that of the other 111-V compounds discussed in this chapter and that the [loo] minima are not the lowest conduction-band minima. Resistivities of the colorless crystals of AIN range from 10'' to 1013R-cm at room t e m p e r a t ~ r e . ' ~ ~ An * ' ~activation ~ ~ ' ~ ~ energy of 1.4 0.1 eV (300" < T < 450°K) was obtained from the temperature dependence of the r e s i s t i ~ i t y , , ' ~which ~ ~ ' ~could ~ be associated with the level responsible for a small increase of optical absorption at 2.8 eV.14' This level also gives rise to photoconduction at 2.8eV and has been shown to be due to interstitial aluminum. 142 Because of the high resistivity, no Hall measurements have been reported. The bluish crystals of AIN (A1,OC) mentioned earlier have resistivities varying between lo3 and lo5 R-cni and from Hall measurements were found to be p-type with Hall mobilities of about 10 cm2 V - ' sec-' at room temp e r a t ~ r e . 'Since ~ ~ these values indicate carrier concentrations of 10" to 1014 ~ m - the ~ ,crystals must be highly compensated.

8. BORONNITRIDE

Cubic boron nitride (BN) was first synthesized in 1957 by W e n t ~ r f ' ~ ~ from boron nitrogen catalyst systems. Later, the compound was obtained by direct transformation from the hexagonal form. 14' The lattice constant is 3.615 A, close to that of diamond with which BN is isoelectronic. Optical data are very meager, since only small crystallites of BN have been prepared to date. No electrical or galvanomagnetic measurements have been reported at all for this material. Philipp and Taft'46 report some rough reflectance data that indicate structure in the region 9 to 10 eV and a peak near 14.5eV, and take these to support evidence of a larger band gap in BN as compared to diamond (5.2 eV). More definitive measurements would be very interesting, since different types of band calculations have yielded quite different values for the band gap. One of these values, from an a priori band calculation using the OPW method,2a is about 1 eV less than that calculated for the diamond band gap. The band gap in BN is expected to be larger than that for diamond since, in general, the gap increases (usually doubles), going from group IV to the 143 '41

J. Edwards, K. Kawabe, G. Stevens, and R. H. Tredgold, Solid-State Commun. 3,99 (1965). R. H. Wentorf, Jr., J . Chem. Phys. 26, 956 (1957). F. P. Bundy and R. H. Wentorf, Jr., J . Chem. Phys. 38, 1144 (1963). H. R. Philipp and E. A. Taft, Phys. Rev. 127, 159 (1962).

1. BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

63

corresponding III-V compounds. Also, the ionicity of BN is relatively large, as evidenced by the electronegativity difference (1.0 eV) and the large difference between the static and high-frequency dielectric constants (c0 - E , = 7.1 - 4.5)147 implying a larger band gap because of the tighter binding between the ions. A larger value for the band gap was calculated by Kleinman and Phillips,'48 who, using a self-consistent crystal potential, applied a first-order perturbation on diamond (the bands of which have also been calculated by the OPW method). Their result, perhaps fortuitous, was 10 eV or about twice the value for diamond. That the value of band gap is smaller in the first calculation mentioned is due principally to a much larger splitting of the conduction states X I and X , , as compared with the decrease in the top valence state obtained going from diamond to boron nitride. The augmented-plane-wave-method (APW) was applied to cubic BN and eigenvalues at 256 different points in the first Brillouin zone were obtained.'49 The general shape and configuration of the energy bands resembled those obtained in the previous calculations mentioned above. The resulting band structure along the principal symmetry directions is shown in Fig. 23. In this calculation, the earlier OPW calculations were used to obtain an estimate of the direct energy gap and thereby to fix the value of the constant potential between the APW spheres, and also, to fix the value of the Madelung correction to the crystal potential which is required because of the large ionic contribution. These two values were then used as adjustable parameters in the theory. Choosing as a value of the direct gap r;, - r;, = 8.8 eV, the authors obtained an indirect gap of 7.2 eV (All three calculations place the minimum gap along the X direction in k space [loo].) A communication by P h i l l i p ~ ' ' and ~ an answer to it by Keown151 discuss the reliability of the APW method as used on BN in light of the reflection structure around 10 eV and the reflectance peak near 14.5 eV which was experimentally found in BN.146 More recently, the pseudopotential method was applied to BN using form factors scaled from ones employed previously in a calculation on BP.' l a The conduction band minima occurred at X and a value of 7.6eV was obtained for the indirect gap. The direct gap at r was calculated to be P. J. Gielisse, S. S. Mitra, J. N. Plendl, R. D. Griffis, L. C. Mansur, R. Marshall, and E. A. Pascoe, Phys. Rev. 155, 1039 (1967). "* L. Kleinman and J. C. Phillips, Phys. Rea. 117,460 (1960). 1 4 9 D. R. Wiff and R . Keown, J . Chem. Phys. 47,3113 (1967). IS' J. C. Phillips, J. Chem. Phys. 48, 5740 (1968). I S ' R. Keown, J. Chem. Phys. 48, 5741 (1968). l5l'L. A. Hemstreet, Bull. Amer. Phys. SOC.15, 1379 (1970). 14'

64

RICHARD J. STIRN 16

I.2

0.8

,$

0.4

w

J

9 v,

I

0.0

fl

a

x5

$j -0.4

x3

z 0 W (r

z

w

-0.8 ~

-I

-1.2

X

r

K

W Z X

U

L

!

FIG.23. Band structure of BN along principal symmetry directions. (After Wiff and K e 0 ~ n . I ~ ~ )

10.7 eV. The latter value could well account for the reflectance structure around 10 eV, which might be expected to be due to direct transitions. 9. BORON PHOSPHIDE

Cubic boron phosphide (BP) has a melting point greater than 3O0O0C, but decomposes into B,P and P, at much lower temperatures. Hence, zone-refined crystals are not attainable. However, single crystals of sufficient size and purity for optical and electrical measurements have been prepared, Methods of growing crystalline BP have been reviewed by Williams.' 5 2 BP has a lattice parameter of 4.537 A and is remarkably inert chemically. It has a hardness comparable to that of SIC. The first optical transmission measurements were made by Stone and on amorphous BP. A large, abrupt decrease in the transmittance 15' 153

F. V. Williams, in Willardson and Goering,' p. 171 B. Stone and D. Hill, Phys. Rev. Lett. 4, 282 (1960).

1.

BAND STRUCTURE AND GALVANOMAGNETIC EFFECTS

1ooo /T,

65

OK-'

FIG.24. Temperature dependence of the Hall mobility of p-type BP. (After Stone and Hill.'s3)

of 6eV was taken to indicate a band gap of that width. However, now it is quite certain that the band-gap energy is only 2 eV. The first report of this lower value was by Archer et ~ 1 . , " ~ who obtained agreement within 50 meV from measurements of optical absorption, injection electroluminescence, and photoelectric response of surface barrier contacts. At about the same time, Wang et a1.'41 reported an absorption edge at 2.0 eV in BP. The low value of the band gap in BP is somewhat surprising when the usual empirical rules are considered. For example, the usual doubling of E , over the value found in the corresponding diamond-type analog (p-Sic, in this case, with a band gap of 2.2 eV) is not followed. An explanation of this fact is apparently that BP is very much covalent, as evidenced by its small electronegativity difference of 0.1 eV. Hence, one has an opposite effect from that of BN with its high ionic character and large band gap. The low ionicity in BP explains the lack of a reststrahlen spe~trum.'~'The covalent nature of BP is also shown by the infrared reflectivity in its vibrational region,14' which suggests an extremely low effective ionic charge. A linear relationship holds when the square root of the absorption R. J. Archer, R. Y. Koyama, E. E. Loebner, and R. C. Lucas, Phys. Rev. Lett. 12, 538 (1964).

66

RICHARD J . STIRN

coefficient in BP is plotted versus photon en erg^.'^^.'^^ This fact and the similarity of the absorption coefficients with those of GaP and S i c suggest that the absorption edge is indirect. The band structure of BP is discussed and compared to the band structure of /?-Sic by Wang et who also associate the structure of the reflection spectrum (maxima at 5.0, 6.9, and 7.9 eV) with direct interband transitions at high-symmetry points in k space (Eo', E , , and El', respectively). A first principles self-consistent OPW calculation by S t ~ k e l has ' ~ ~very recently been published and, despite a lack of complete convergence (459 OPW's used at r),appears to give reasonable results. The calculation confirms the experimental measurements indicating a band gap of 2 eV and places the conduction band minimum along the A line 0.81 of the distance from r to X rather than at X itself. Stukel suggests that the 6.9eV main reflectivity peak'41 is due to transitions in the outer part of the zone (U-K region) rather than due to the X-point transition. He also suggests that the 5.0eV weak reflectivity maximum should not be assigned to I-;, - r;, (EO'),l4'but rather to the L,' - L," transition. Electrical measurements on crystals of BP show that most samples are p-type, although needle-shaped crystals that are n-type have also been grown.'53 Stone and Hill'53 measured carrier concentrations of 1-5 x 10" cmV3in their p-type crystals, which showed a constant Hall coefficient from about 900" down to about 160°K. The shape of the resistivity curve at low temperatures suggested that their material was still in the exhaustion range at 78°K. The Hall mobility for holes as a function of temperature is shown in Fig. 24. It is seen that impurity scattering is dominant even at room temperature. Crystals grown by Wang et ~ 1 . ' ~were ' apparently more pure as hole mobilities up to 500 cm2 V-' sec-' at room temperature were obtained. The resistivity wasabout lo-' R-cm at 300"K, while Hall measurements showed that the carrier concentration was 10" cm-3 at 300"K, and l O " ~ r n - ~at 78°K. ACKNOWLEDGMENTS I would like to express my appreciation to Drs. W . M . Whitney and W . M . Becker for reading this work and for helpful suggestions, and to Dr. F. Herman for stimulating conversations and for sending me unpublished reports. It is also a pleasure to express my indebtedness to Miss M. Brandenberg for her careful typing of the manuscript.

155

D. J. Stukel, Phys. Rev. B1,4791 (1970).