R 391 Philips Res. Repts 15, 107-119, 1960

DETERMINATION OF NUMBERS OF INJECTED HOLESAND ELECTRONS IN SEMICONDUCTORS

by F. van der MAESEN 537.311.33

SummaryIn semiconductors, deviations LlIl and Lip of thè equilibrium numbers110 and po of electrons and holes are unequal in many cases because oftrapping. It is shown how measurements of the photo Hall-effect andphotoconduction may give information on the numbers LlIl and Lipseparately. From the ratio Y of the relative change of the Hall-effect

C and of the-photoconductivity, the quantity K = LlIl/Lip can be evaluated.Graphs of Y versus K for some 'substances are givenült is shown howK occurs in the formulae used in computations of the diffusion-re-combination length L from photoelectromagnetic and photoconductivedata. If r« and r» denote the ratios of the Hall-mobility to drift mobilityfor electrons and holes, respectively, rn/rp can be evaluated from themeasurements where K = 1. In this way the value of rn/rp at room tem-perature is found to be 0·54 ± 0·05 for germanium and 0·78 ± 0·04 forsilicon.

RésuméDans les semi-conducteurs les déviations LlIl et Lip des quantités respec-tives d'électrons 110 et de trous po existant en équilibre sont souvent dif-férents par suite des phénomènes de capture. On montre que les mesuresde l'effet de Hall photoélectriques associées aux mesures de photocon-ductibilité dorment des renseignements séparément sur les nombresisolés LlIl et Lip. On trouve Ie rapport K = Anl áp en partant du rapportYentre les déviations relatives de l'effet de Hall et de la photoconducti-bilité. On donne pour quelques substances les diagrammes de Y enfunction de K. On montre comment Ie rapport K entre dans les formulespour le calcul de la longueur de diffusion en partant des effets photo-

, électromagnétiques et de la photoconductibilité. En appelant rn et rples rapports de la mobilité de Hall et de la mobilité de dérive respective-ment pour les électrons et les trous, Ie rapport rn/rp peut être calculépar les mesures à partir des mesures dans Ie cas oü K = 1. De cettefaçon on peut trouver à la température ambiante la valeur r-Jr» = 0,54± 0,05 pour le germanium et rn/rp = 0,78 ± 0,04 pour le silicium.

ZusammenfassungIn Halbleitern sind infolge von Einfangprozessen oftmals die Abwei-chungen LlIl und Lip der im Gleichgewicht vorhandenen Anzahl von Elek-tronen 110 bzw. von Löchern Po pro Volumeinheit ungleich. In dieserArbeit wird diskutiert, wie man durch Messungen des Photo-Hall-Effekts in Verbinding mit Messungen der Photo-Leitfähigkeit Infor-mationen über die Werte von LlIl und Lip gewinnen kann. Aus dem Ver-hältnis Y der relativen Änderung des Hall-Effekts zu der der Leitfähig-keit läl3t sich das Verhältnis K = LlIl/Lip ermitteln. Für einîge Stoffewerden Diagramme von Y in Abhängigkeit von K gegeben.' Es wirdgezeigt, wie das Verhältnis K in die Formeln für die Berechnung derDiffusionslänge aus Daten des photoelektromagnetischen Effekts undder Photo-Leitfähigkeit eingeht, Das Verhältnis von Hall-' zu Drift-beweglichkeit sei r« für Elektronen und r» für Löcher genanrtt. Aus,Messungen in Fällen von K = 1 kann das Verhältnis rn/rp berechnetwerden. Auf diese Weise wird für Raumtemperatur der Wert r-lr» -= ,0,54 ± 0,04 für Germanium und rn/rp = 0,78 ± 0,04 für Silizium ge-> \\funden.

108 F. van der MAESEN

Introduetion

In cases of injection by light or otherwise one often assumes an equalityof numbers of extra mobile holes and electrons LJp and LJn. This is justified inmany cases in high-purity germanium and silicon, but certainly not in the semi-conductor compounds available at present, where a large number of traps mayoccur. Several authors have already discussed the influence oftrapping on photo-conduction and photoelectromagnetic effects 1) 2). In the paper by Zitter 2) it isshown how, in the case of trapping, different lifetimes 'TPEM and 'Tpc can bederived from photoelectric and photoconductivity measurements, provided thephoton-flux density ofthe exciting light is known and the surface-recombinationvelocity is neglected. From these lifetimes the fraction of excess carriers whichare trapped can be evaluated. In this paper we shall discuss the photo -Hall-effect which - in combination with photoconduction - may be used in orderto obtain the necessary information on the numbers LJn and LJp and consequentlyon trapping.

Rnt = ant {Nt (1 - fi) n - Nt fi nH},Rpt = apt {Nt fip - (1 -fi) Nt pH}.

(1)

(2)

1. General considerations

Suppose we consider a semiconductor containing s sets of recombinationcentres with densities N; (i = 1,2, ... , s). According to formulae first given byShockley and Read we may write for the capture rates of electrons and holesby the ith set:

In these formulae ant an~ apt are recombination constants (cross-section 'xthermal velocity) for electrons and holes, fi is the fraction of N; centres whichare occupied by electrons, pand n are the total numbers of holes and electronsand PH and nH are the numbers of holes and electrons when the Fermi levellies at the energy level of the ith set of centres. For very small deviations LJn,LJp and LJj:"from equilibrium values, (1) and (2) may be written as

Rnt = ant {Nt (1- fio) LJn - (no + nH) N; LJfi},Rpt = apt {Nt fio LJp + (po + pH) N, LJ fi} ,

(3)

(4)

where the subscript zero refers to normal equilibrium conditions. If we have ap-type sample, the linear expansion holds as long as LJn « no + nH.Formulae similar to (3) and (4) are also used by Wertheim 3) in his article onthe transient recombination of excess carriers in semiconductors.We consider now a situation in which small stationary deviatieris LJn and LJp

are introduced by injection from a p-n junction or by generation with light.We may write down the steady-state condition for each set of centres:

Rnt = Rp! (i = 1,2, ... , s), (5)

DETERMINATION OF NUMBERS OF INJECfED HOLES AND ELECTRONS IN SEMICONDUCfORS -109

which implies that there will be no change of the occupation of the centresdue to the action of light if present; and also that the transfer of charge betweenthe sets of centres is neglected. In addition to the s equations of (5), an equationis obtained from the equation of Poisson, relating the gradient of the electricfield and the space charge: .

sdE/dx = (q/€O€) {Jp - Jn - ~ Ni Jfi}.

1=1

(6)

Investigation shows that the local excess of charge required to build up anelectric field if gradients are present is usually very small. When we confineour considerations to those parts of the semiconductor which do not containspace-charge regions such as occur in p-n junctions or which are not adjacentto those regions, we may introduce the approximation

sJp-Jn-~Ni Jfi = o.

1=1(7)

This eq. (7) is often referred to as the condition of space-charge neutrality. Itshould be noted that the use of (7) does not mean that the field gradient in (6)also vanishes. Equations (5) and (7) give s + 1 linear equations for s+ 2variables Jn, áp, Jfl, ... , Jfs. Consequently the ratios between these variablesare constant, i.e.,

ánj áp = K. (8)

Under the conditions mentioned (small deviations from equilibrium, steady-state conditions, space-charge neutrality, no action of light on centres and notransfer of charges between centres), K will be a constant independent of timeand position. .Using (3) and (4), eq. (5) may be written as

Rni = Rp! = f3i Jn + Yi áp , (9)

where f3i and Yi are constants. Hence we find for the total capture rates forholes and electrons a similar expression:

s s

u; = ~ Rni, Rp = ~ Rp!, !1=1 1=1

Rn = Rp = f3Jn + yJp.

(10)

The continuity equations for the holes and electrons may be written as

1- 17 Jn-Rn + gn(X,y,Z) = 0, .q .

1-- 17 Jp-Rp + gp (x,y, z) = 0,

q.

(11)

(12)

110 F. van der MAESEN

where gn and gp in (11) and (12) are the generation terms. As the divergence ofthe total current vanishes, it follows from the equality of the recombinationrates that the desired steady-state situation can only be achieved if at all pointsin the semiconductor gn = gp. This means that, if there is a generation, it mustbe in hole and electron pairs. For gn =gp = constant the solutions for Lln and LIpof (11) and (12), under the conditions that (8) and (10) are valid, are simpleexponential functions.For non-stationary cases the situation is more complicated 3)4) and K as

defined in (8) will generally be a function of time and position.

2. Photoconductivity and photo Hall-effect

Let us consider a flat sample of a p-type semiconductor withdimensions large'compared to the thickness d and the diffusion length L;' of the minority carriers;see fig. 1. The sample is provided with suitable Hall- and resistivity contacts;the Hall-contacts run along the whole thickness d.

------'8"Front

I

Fig. 1. Geometry and experimental configuration of p-type sample.A and B current contacts, C and D contacts for resistance and photoconductivity measure-ments, E and F contacts for Hall and photo Hall-measurements.The current contacts are supposed to' be extended over the whole cross-section.

When this sample is illuminated uniformly at one side, it is possible to mea-sure the relative change in conductivity, due to illumination, as the. relativechange in' voltage on the resistivity contacts C and D under conditions ofconstant current:

(LIVe/Vc)! = - (L1p/po) - b (L1n/po). (13)

Here L1n and L1p are the numbers of extra mobile electrons and holes, averaged

(L1VH) L1p (rn ) L1n--, =--- -b2+2b -.-'VH I PO r» PO

DETERMINATION OF NUMBERS OF INJECTED HOLES ANI? ELECTRONS IN SEMICONDUCTORS 111

across the thickness d, and their density is assumed to be small compared withthe density po of the majority carriers; b = p.n/ p.p is the ratio of the electron- tohole-mobility.The relative change of the Hall-voltage, measured at the contacts E and F

in a suitable magnetic field Ba; 'in the x-direction under the same conditions ofillumination and constant current is

(14)

This formula is easily derived by logarithmic differentiation of the usual for-mula of the Hall-constant for two types of carriers. Here r« and rp are the ratiosof the Hall-mobility to drift-mobility for the electrons and holes respectively.Derivations which lead to (13) and (14) are discussed in the appendix.From measurements of both effects Anlp« and L1plp« can be deduced separa-

tely. With respect to these relative quantities the formulas (13) and (14) areof quite general value. For instance, beside stationary measurements, transientmeasurements of (13) and (14) may be carried out, giving L1n/po and L1p/po asseparate functions of time.As mentioned in sec. 1, the conditions for a constant ratio L1n/L1p = K, .

independent of time and position, can be realized using light with sufficientphoton energy to create hole and electron pairs: gn = gp. This light is absorbed.within a very thin region of the surface, whereas farther inside the semicon-ductor gn = gp = O.In those cases K may be computed from

Yp = (L1VH) j(L1Vc) = l/K + [(rn/rp) b2 + 2b], (15)

VH I Vc I 1/K + b

provided values for rn, rp and b are available. It should be noted that in (13)and (14) only the varages L1n and 4p in the x-direction occur. Therefore Kevaluated from (15) is independent of the distribution of the extra carriers inthe x-direction. The corresponding formulae for the photoconductivity andphoto Hall-effect in n-type samples are

(16)

(17)

'Yn = (L1VH) j(L1Vc) ..= K + [(rp/rnb2) + 2/b],

VH I Vc I K+l/b(18)

where no is the majority-carrier equilibrium density in the n-type sample.

13 Yp=[~~J/t~Jfor p-type crystals'21- HIe I ~f-

11 1//

0/

8 ijsc6f- r-~p t-S

1/

" J 0"/.~c

3 1/ ,('I

2V ~ ...lL ~"'~

f-'/ 9

,V ~ ,./'

0-2 a 3 5 -1 a 3 5. a s 3 510 10 10

112 F. van der MAESEN

Figures 2a and 2b give the graphs of the ratios Yp and Yn respectively as afunction of K, appropriate for germanium, silicon and silicon carbide. Thevalues of band I'n/I'p are taken as follows:

GermaniumSilicon

b= 2,b = 5,

I'n/I'p = 0·54 *),I'n/I'p = 0·78 *),I'n/I'p = 1.Silicon carbide b = 10,

Fig. 2a. Ratio Yp as defined in eq. (I5) as a function of K.

From the Yp-curves in fig. 2a we see that it will be possible to determinethe trapping of minority carriers in the p-type crystals with sufficient accuracy.The same holds for trapping of majority carriers in n-type samples, as can beseen from fig. 2b.Accurate measurements of the ratio Ymake it possible to determine the ratio

I'n/I'p in cases where K and b are known. This will be shown for germanium andsilicon in sec. 3.Another application of the photo Hall-effect and the determination of K lies

in the interpretation of measurements of the photoelectromagnetic effect andthe photoconductivity, which are usually carried out in order to determine thediffusion length L of the minority carriers. If we assume that the distributions

*) See sec. 3.

Lln = An exp (- xfLn), LIp = Ap exp(-xfLn), (19)

DETERMINATION OF NUMBERS OF INJECTED HOLES AND ELECTRONS IN SEMICONDUCTORS 113

of extra carriers under illumination in the p-sample of fig. 1 are given by theexponential functions (see fig. 3)

3

l

A· 1~v.] I _~v._1 11 1 1 11..1:~"- Jó =t ~/~l for n- type crystals-

r-...8 -,.6 1\.4

"~

~ I~\.0,1\"

'3

r"\ 1\.8

'~\

6 ~% r\~

1\~ ..,.4 '1ó

1\I'-~ t-,

i'--..0

8-22 3 5 -123 5 a 3 5 310 10

_K

Yn

f :

IJ.

1.

o.10

USO?

Fig. 2b. Ratio Yn as defined in èq, (18) as a function of K.

z

lightC A A -x/Ln

""p= pe

11588

Fig. 3. Distribution of extra carriers in the illuminated sample in the x-direction, K < 1

114 F, van der MAESEN

we find for the linear photoelectromagnetic voltage VPEM = Vo-VD, developedunder the influence of a magnetic field By in the y-direction 5),

VPEM .;_ (kT/q) (rp + brn) J.tn':ByhAn/dpo , . (20)

In this formula An represents the extra.electron concentration at the illuminatedfront surface (x = 0), po the majority-carrier equilibrium density, h the dis-tance between the contacts C and D, and d the thickness of the sample, Theexpression (20) is valid for small injections and where Ln <:< d. The quantity Andepends on the photo-flux density, the diffusion length and the surface-recom-bination velocity,According to (13) the photoconductive effect under the same conditions of

illumination is in absolute value

(LIVe) = (2_ + b) Ln An ,Vc IK. d PO

Measuring the effects (20) and (21) and taking their ratio leads to

kT/q rp + brn (LIVc)Ln = --- -- J.tnByh.VPE1II l/K + b Ve I

The corresponding formulae for injection in an n-type sample are

(21)

(22)

kT ( rp) h ApVPEM = - rn + - J.tpBy--,q b d no

(23)

(LIVe)' = (K+~) Lp Ap,Vc I b d no

(24)

kT/q rn + rp/b (LIVc)Lp = -- - J.tpByh.VPEM K + l/b Vc I

(25)

From the formulae (22) and (25) we see that the knowledge of K is required incomputing L from p.e.m. and p.c. measurements. An additional measurementof the photo Hall-effect may give the necessary information to derive a correctvalue for L.When the contacts are suitably arranged (see fig. I), measurements ..of the photoconductivity, p.e.m. effect and photo Hall-effect are all easilypossible on the same sample, provided that the sample and the direction of thelight can be given the two necessary positions with respect to the magnetic field.

In more-complicated cases the determination of K.or the numbers of extraholes and electrons separately may serve as an indication of the trapping ofone of the types of carrier in traps or recombination centres. Beside trapping,other causes of a difference in the numbers Lln and LIp may exist and may befound by a determination of K. An example of this possibility is the extraaccumulation of one type of carrier in the surface due to illumination which

DETERMINATION OF NUMBERS OF INJÈcrED HOLES AND ELECTRONS IN SEMICONDUCTORS 115

causes extra surface conductivity. If values of K = 1 have to be attributed tothis phenomenon rather than to trapping, one may expect different K-values forvarious applied surface treatments. .

3. Measurements on germanium, silicon and sllicon-carblde crystals

Measurements of the photoconductivity and photo Hall-effect at room tem-perature were performed on flat p-type germanium, silicon and silicon-carbidesamples. Contacts were made along the whole thickness of the samples, and formeasurements the crystals were placed between the poles of an electromagnet,capable of giving a field of.5000 oersteds. The samples were illuminated at oneflat side with the aid of a mirror, mounted in a suitable 45° position betweenthe poles of the magnet. In this way photoconductivity and photo Hall-effectmeasurements could be carried out easily by switching on and off the light.Merely by turning the sample over 90°, it was possible to perform measurementsof the photoelectromagnetic effect and corresponding photoconductivity, ifdesired. In the experiments white light filtered by a water layer was used for thegermanium and silicon, and ultra violet from a HPW 125 W for the siliconcarbide. Under these conditions the requirement for the generation of holes andelectrons in pairs may be assumed to be fulfilled.The measurements were taken under steady-state conditions of illumination

and the small photoconductive and photo Hall-effects were determined with aDiesselhorst compensator and a Philips GM 6010 electronic voltmeter as in-dicator. As an illustration of the discussions in the preceding section, table Igives a summary of characteristics of the samples together with some measure-ments. It should be noted that the figures in the table for the photoconductivityand the photo Hall-effect were obtained for a current of 1 mA and a certainunknown intensity of illumination.

The germanium sample measured J X 1XO'3 cm" and was cut from a highlypurified crystal, prepared in our laboratory by J. Goorissen. The sample wasetched in a CP4 solution, then sandblasted, with the exception of the frontsurface. The K-value may be very well taken as 1in this case, which, as discussedin the preceding section, leads to the possibility of determining rn/rp.

For various currents and intensities of illumination we found the ratio Yp

to be 2·38 ± 3%, which leads to a value of I'n/I'p = 0·54± 0·05 for germaniumat room temperature. The best values reported in literature are r« = 1, I'p =1.8 at room temperature 6), leading to rn/I'p .::_0·55.There was no change in thevalue of Yp when the surface conditions were varied by changing the gas at-mosphere from damp nitrogen to dry nitrogen with a few per cent ozone, whichmeans that the influence of surface-conductivity changes due to the illuminationmay be neglected in this case. This is supported by the fact that photoelectricmeasurements resulted in all cases in the same value of diffusions length Ln =0·11 cm for the electrons.

116 F. van der MAESEN

TABLE I. Results of measurements on p-type germanium, silicon and silicon-carbide crystals

quantity I symbol I p-Ge I p-Si I p-SiC

specific resistivity, .Q cm pp 12·65 85·6 422mobility, cm2/Vsec fJ-p 2258 344 26majority-carrier density, cm-3 po 2.1.1014 5,7.1013 5.7.1014mobility ratio b 2·0 5 10photoconductivity LJVc/Vc 0·00971 0·0185. 0·024photo Hall-effect LJVH/VH 0·0229 . 0·0965 0·050ratio ofeffects Yp 2·36 5·20 ' 2·08ratio extra carrier densities fÈ 1 1 0·011relative number extra electrons LJn/po 0·0032 0·0031 2.4.10-4relative number extra holes LJp/po 0·0032 0·0031 2.2.10-2ratio Hall-effect factors

~0'53 0·81 1

The silicon sample was a disk with a diameter of 1·4cm and a thickness of0·1 cm, cut from a crystal prepared in our laboratory by J. Goorissen. Like thegermanium crystal the sample was etched and then sandblasted, except thefront surface. Ifwe take the K-value as 1, the averagevalue for Yp of5·10 ± 3%leads to a value of rn/rp = 0·78 ± 0·04 for the silicon at room temperature.The silicon-carbide crystal was prepared in our laboratory according to the

method of Lely 7) by W. F. Knippenberg. The sample had an arbitrary shapewith dimensions of'O-S cm and a thickness ofO·06cm. No surface treatments wereapplied. The measurements of the resistivity and Hall-effect on this sample werecarried out according to the side-contact method of Van der Pauw 8). For theanalysis of the measurements on this SiC sample the Yp-curve for silicon carbidein fig. 2 was used (rn/rp - 1), resulting in a K-value of about 0·01. This meansthat almost all of the injected electrons are trapped. A similar situation seemsto occur in the high-ohmic n-type silicon-carbide layers described by Patriekand Choyke 9). In accordance with this information is the fact that it was im-possible to detect any photoelectromagnetic effect despite the good photo-conductivity.

Acknowledgement

The author wishes to express his gratitude to Mr F. H. Stieltjes and ProfessorPolder for valuable discussions and to Misses A. B. Crajé and A. de Bruin forcarefully carrying out preparations and measurements.

Eindhoven, September 1959

Jp = -apV CPp- ap(V CPpX B),

Jn = -anV CPn+ an(V CPnX B).

CPnand CPpare the "quasi Fermi-potentials" as defined by Shockleyt''):

(Al)

(A2)

DETERMINATION OF NUMBERS OF INJECTED HOLES AND ELECTRONS IN SEMICONDUCTORS 117

Appendix

If we neglect the quadratic terms in the magnetic field, we may write downthe general relations which give the current densities of holes and electro~s inthe presence of an electric and magnetic field in a semi-conductor where extra:carriers are present:

CPp= (kT/q) In (p/pi) + ifJ, (A3)

(A4)CPn= - (kT/q) In (n/ni) + ifJ.

Pi = tu in (A3) and (A4) are the intrinsic numbers of carriers at the given tem-perature; ifJ is the electrostatic potential. The coefficients in (AI) and (A2) areall positive and defined by

(A5)

(A6)

where pand n are the total densities of holes and electrons. The other symbolshave their usual meanings.We restrict ourselves now to the situation of fig. 1. For infinite dimensions

of the semiconductor in the y-and z-directions we may assume the gradientsb/by and b/bz of the number of carriers and the electric field to be zero. Thisideal situation is approximated experimentally by taking a :flat sample. withdimensions in the y- and z-directions that are large with respect to the thicknessand the diffusion length of the minority carriers.

From (AI) and (A2) we deduce for the components of the total currentdensity J in the case of illumination

Ix = aEx - (kT/q) b(ap -:- an)/bx,

Jv = «s, + aEzBx,

Je = aEz - aEyBx,

(A7)

(A8)

(A9}

where a = ap + an, a = ap - an, E = - V ifJ.Appropriate solutions for the electric-field components are

kT I b(ap - an)Ex=-- ,q.a . öx

Ey = Cl, s, = C2,

(AIO)

(All)

where Cl and C2 are constants; Ex in (AIO) is the Dember-field.

118 F. van der MAESEN

Having regard to the experimental conditions we may write for the averagesin the x-direction of Jy and Je:

Jy = äEy + äEzBx = 0,

Jz = äEz - äEyBx.

(AI2)

(A13)We introduce now

ä = ao + Li~, ä = ao + Lla, (AI4)with

ao = qfLppo + qfLnno, Lla = qfLpLip + qfLnLln, (AI5)

ao = I"pqfLp2po- I"nqfLn2no, . Lla -: I"pqfLp2Lip - I"nqfLn2L1n. (AI6)

The subscript zero refers to normal conditions without illumination.Considering the situation at the measurement of the photoconductivity we

have Jz = Jzo, B« = °andäEz = aoEzo,

which leads to

(LIVe) _Ez-Ezo_ Aa _ Lla- _ _--------=-,Ve I Ezo ä ao+ LIa

(AI7)

which is the correct expression for the relative photoconductivity. For smallextra carrier densities with respect to the majority-carrier density, (AI7) leadsto (13) and (16).In the measurements of the photo Hall-effect we have Jz = Je«, Jy =

Jyo = 0. Consequently

äEz - äEyBx = aoEzO- aoEyoBx ,

äEy + äEzBx = 0,

aoEyo + aoEzoBx = 0 .

Still neglecting terms of the second degree in B one can derive from theseequations

(AI8)

which equation for small deviations from equilibrium reduces to

(LlVH/VH)I = (Lla/ao) - 2(Lla/ao) (AI9)

leading to (14) and (17) respectively.The importance of the formulae given for the photoconductivity and photo-

Hall-effect is that they are valid for any distribution of extra carriers in the x-direction such as occurs UpOl1illumination with non-penetraing or partlypenetrating light. Itmay be assumed that the formulae hold for thin flat samplesof arbitrary shape.

DETERMINATION OF NUMBERS OF INJECTED HOLES AND ELECTRONS IN SEMICONDUCTORS 119

REFERENCES

1) A. Rose, Proceedings of the conference on photoconductivity, Atlantic City, Nov. 4-6,1954 (John Wiley and Sons, Inc., New York, 1956)p. 17.

2) R. N. Zitter, Phys, Rev. 112, 852-855, 1958.3) G. K.Wertheim, Phys. Rev. 109,1086-1091,1958.4) K. C. Nomura and J. S. Blakemore, Phys. Rev. 112, 1607-1615, 1958.5) W. van Roosbroeck, Phys. Rev. 101, 1713-1725,1956.6) F. J. Morin, Phys. Rev. 93, 62-63, 1954.7) J. A. Le ly, Ber. dtsch. keram. Ges. 32, 229-231, 1955.8) L. J. van der Pauw, Philips Res. Repts 13, 1-9, 1958.9) L. Patrick and W. J. Ch oyke, J. appl. Phys. 30, 236-248,1959.10) W. Shockley, Electrons and holes in semiconductors, Van Nostrand Co., New York,

1951, p. 308.