R 767 Philips Res. Repts 26, 279-297,1971

FOUR-POINT-PROBE RESISTIVITY MEASUREMENTSON SILICON HETEROTYPE EPITAXIAL LAYERS

WITH ALTERED PROBE ORDER

by P. J. SEVERIN

Abstract

The four-point-probe method for resistivity measurements is applied toa thin, epitaxially grown, hetero type silicon layer with an imperfectlyisolating junction to the well-conducting substrate. A bar-shaped struc-ture is analysed for low-voltage operation, < k'F[e, and for high-voltageoperation as far as analytically possible. The more usual geometry oflaterally infinite extent is dealt with in the low-voltage range and it isshown that both the sheet resistance and the interface zero-bias resistancecan be found from two four-point-probe measurements with alteredprobe order. The theory is substantiated with semi-quantitative experi-ments. It is advocated that four-point-probe measurements are done atmillivolt level. The precision and accuracy of four-point-probe meas-urements are discussed.

1. Introduetion

Among the various methods available for measuring characteristic parametersof semiconductor material, in particular silicon, the four-point-probe resistivitymeasurement is generally considered to be the most indisputable one.

The main advantage of the instrument is that current and voltage are read intwo well-separated circuits which ideally are coupled by the sample only. Themain disadvantage is the large integration surface area amounting to severalmm" with normal probe distance s = 660 fL. The precision and accuracy whichcan be attained with the four-point-probe method are determined by the uni-formity of the sample, the rigidity ofthe instrument on the one hand and by theavailability and applicability of a suitable theory on the other hand.

In modern semiconductor technclogyg epitaxially grown silicon layers arerequested of an increasingly more uniform and better specified sheet resistance.This demand can be met only if the evaluation methods also do allow for suchaccuracy and precision. This applies particularly to as elegant and rapid amethod as the four-point-probe one. Therefore, the author considers it approp-riate at this stage to present a fairly comprehensive review of the literature onthe four-point-probe method. It will be found that very little work has beendone on structures dominated by conditions prevailing with thin, heterotype,epitaxially grown layers with imperfectly isolating interface to a well-conductingsubstrate. The aim of this paper is to elaborate the classical theory as needed forthe interpretation of measurements on such structures.

The original theory of four-point-probe measurements has been given by

280 P. J. SEVERIN

Valdes 1). He states explicitly a number of conditions which should be satisfiedfor reliable measurements. The most important one is that the surface shouldhave a high recombination rate so that minority-carrier injection is ineffective.He also works out by the method of images the corrections to be applied forprobes perpendicular or parallel to a conducting or non-conducting boundaryand for a thin slice with conducting or non-conducting bottom surface. This wasfurther evaluated by Vhlir 2). The correction factors for a thin slice of finitedimensions were calculated, also by the method of images, by Smits 3) andLaplume 4). Correction factors for an arbitrary arrangement of four-pointprobes on the edges of a thin slice are given by Van der Pauw 5), for an in-linefour-point probe perpendicular to the radius of a finite thin slice by Albert andCombs 6) and Combs and Albert 7), for bars offinite radius by Gegenwarth S),for samples of infinite length and with rectangular or semi-circular cross-sectionby Hansen 9), and for cylindrical samples by Murashima and Ishibashi 10).Dependence on measuring current level is discussed by Garrison 11). A generaldiscussion of various problems regarding geometrical corrections and currentlevel is given by Gutberlet-Vieweg and Schönhofer 12).The accuracy of four-point-probe measurements is discussed by Hargreaves

and Millard 13). They mention heating, minority-carrier injection and probewander as possible sources of error. They derived an expression for the fractionaldecrease in resistivity due to minority-carrier injection at the forward-biasedcontact.Automated instruments have been described by Dew-Hughes et al. 14) with a

lifted probe, and by Brice and Stride 15) with a rolling probe. Different construe-tions of probe heads are described by Kennedy 16), Paulnack and Chaplin 17),and Clerx IS), who advocates the use of hardened-steel alloy or unplatedtungsten carbide as probe materials. Steel generally has a lower contact resist-ance to silicon than tungsten carbide. Tong and Dupnock 19) report on thedependence of four-point-probe measurements on probe-loading and tip radiusof curvature: a broad-contact probe reproduces better, because a tip of smallerradius might penetrate the layer and produce erratic readings.A special arrangement for circular slices is given by Gergely and Hantay 20)

and Bernikov and Rvachev 21).Various measuring circuits have been suggested for eliminating thermal poten-

tials and rectification of spurious signals at reversely biased probes. Dauphinéeand Mooser 22) developed a d.c. sampling technique. A.c. measuring circuitshave been discussed by Logan 23), Allen and Runyan 24), Barry and Edwards 25)and Tarui 26). Mercury contacts, pressed against the specimen surface have beenreported by Cooper and Lerner 27) to yield satisfactory results on Mo permalloyand CoP layers. They remark that this system can be used for semiconductorsalso: Ge is slightly soluble in Hg, and Si does not amalgamate at all.The influence of non-uniform conductivity on in-line- arid square-four-point-

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 281

probe measurements on thin finite-size slicesis calculated by Swartzendruber 28).Measurements on thick slices of infinite extent and anisotropic conductivity withan in-line-probe arrangement are described briefly by Airapetyants andBresier 29). This is elaborated more fully with thin slices and square-probearrangements by Wasscher 30). A direct-reading instrument for 3.10-3- to103_ 0 cm-Si four-point-probe resistivity measurement is described by Swart-zendruber et al. 31). A square four-point probe has been discussed by Keywelland Doroshevski 32) with correction factors for the finite size of a square sample,and of a circular sample by Vaugham 33). Zrudsky et al. 33) advocate inter-changing the voltage and current probes of a square-four-point-probe arrange-ment because after averaging the two readings the result depends on misalign-ment ofthe probes to second order only. The solutions for different symmetricalconfigurations of four-point probes, in particular for a diamond-shapedarrangement, and the result of different probe order are calculated byRymaszewski 35).A detailed description of the four-point-probe procedure and instrument

requirements are given in the ASTM proposal 36), which should make themethod suitable for resistivity measurements of 5.10-3 - 120-0 cm Si with aprecision of ± 2% at 23°C. Heating increases the resistivity, and temperature-coefficient vs resistivity (! curves are presented by Bullis et al. 37) for Si and Geat 23°C. They can be used for specimens doped with shallow-level impurities. Therelative temperature coefficient is smaller than 1%tc over the whole resistivityrange. Wilmanns 38) remarks that when the dependence of (! on the temperatureis known, measurement of the sheet resistances of a thin slice at two tempera-tures yields simply the thickness of the layer.Valdes 1) did not recommend his method for a very thin layer on a conducting

substrate. This is certainly valid for an n-on-n+ -type Si epitaxially grown layerwith a probe separation s = 660 (Lm and a thickness d of several (Lm. Differentembodiments have been published to overcome this difficulty: the over-underfour-point probe by Schumann and Rallenback 39), Schumann and Sheiner 40)and Gursakov et al. 41) and the delta four-point probe by Schumann andGardner 42). A five-point in-line probe is described by Hora 43). It is designedfor measuring the resistivity (! and the thickness d of a layer on an ideally ornon-conducting substrate. The probe holder is a normal in-line four-point probea constant distance s apart and a fifth s* apart from the fourth. The correctionfactors to be applied when using the normal four probes Ol the fifth instead ofthe fourth are computed and from both readings (! and d can be calculated.The most commonly chosen solution to the practical problem of characteriz-

ing an n-on-n+ epitaxiallayer is to measure the sheet resistance on a check sliceconsisting of a simultaneously grown n-type layer on a p-type substrate. Thetransition between the layer and the substrate is considered to be sufficientlyisolating to warrant the use of Valdes' asymptotic expression for four-point-

282 P. J. SEVERIN

\;1'probe measurement of a thin layer on an isolating substrate. Such an arrange-ment should yield a linear current-voltage relationship which, as Patriek 44) hasshown experimentally, does not apply beyond a certain current level. Hediscussed this observation qualitatively as being due to substrate shunting anddepletion-layer expansion. Tong and Dupnock 19) noticed a two-level effect inthe current-voltage relationship with a transition at about 100mY, the lowersheet-resistance level being caused by shunting via the substrate at highercurrent level.It is the purpose of the present paper to investigate theoretically and experi-

. mentally the effects of a non-ideal isolation between the epitaxiallayer and thesubstrate. In sec. 2.1 this junction conductance is supposed to be ohmic, whichapplies either to bad epitaxy or to low-voltage operation « kT/e). In a well-controlled epitaxial deposition process the slices produced show a diodicjunction conductance at higher potentials, the effects of which are investigatedin sec. 2.2. For mathematical convenience the problem will first be treated in a .linear geometry and solved analytically as far as possible; subsequently the morerealistic infinite geometry will be used. Then the problem can be solved onlyunder the linear, low-voltage conditions and it turns out that from a singlemeasurement the layer-sheet resistance e/d and the junction resistance R; can-not be found separately. However, when the measurement is done again withaltered probe order, these two parameters can be found separately. This will bethe subject of sec. 2.3.These considerations are not only applicable to epitaxially grown layers, but

also to diffused layers which can also be investigated with the four-point probe.However, in this paper only layers which have a uniform distribution of dope,both horizontally and in depth, will be dealt with.The theory will be substantiated semi-quantitatively for the low-voltage range

by experiments to be presented in sec. 3. Numerical agreement cannot beobtained in a definite way because many heterotype epitaxiallayer systems turnout to have a non-uniform zero-bias junction resistance or a uniform zero-biasresistance shunted with non-uniform ohmic resistance. The measurement resultsalso turn out to depend strongly on the type ofprobe head used. More definiteand.quantitative measurement results will be presented in a subsequent paper?").

2. The potential distribution in a thin sheet

2.1. Linear geometry, low voltage

The system, shown in fig. 1, will be analysed for simplicity in a bar-shapedgeometry of width B and length L. The layer resistivity and thickness are e andd«L and the substrate is supposed to be of infinite extent and conductivity.Then, as can be seen from the equivalent circuit in fig. 2, the voltage drop in thex-direction is given by

d- [V(X)- Vb] = -Ro i(x),dx

and the current density J, in the z-direction by

di(x)-=-BJz(x),dx

(1)

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 283------

Fig. I. Geometry for four-point-probe measurements on a thin layer.

!.0l. V(x) Ra

--:EfT_x l1,

Fig. 2. Equivalent circuit for the linear geometry model; apart from a factor (2 nr)-l it isalso valid for the infinite geometry around a probe.

(2)

'where the layer resistance per unit length is Ro = e/Bd and Vb is the potentialofthe equipotential substrate.

Further the junction characteristic is supposed to be uniform and equal to

(3)

Equations (1), (2) and (3) can be combined to yield the differential equation

d2V 1---(V- Vb)=Odx2 A2 '

(4)

where the characteristic length A = (Rc/RoB)1/2 = (Rcd/e)1/2.When the substrate surface is not connected to the circuit the floating poten-

tial Vbmust be equal to half the applied potential Vo for symmetry reasons.From the two boundary conditions V= Voat x = 0 and V= 0 at x =L thesolution is found to be

Vo { exp( -x/A)- exp [(x- L)/A]}V=- 1+ .

2 1- exp(-L/A)

Two extreme situations can easily be discussed: L » A and L «A, for whichthe potential distributions are found as

(5)

284 P. J. SEVERIN

Vo ( -x X-L)V = 2 1+ exp11- exp ----:;::! , (6)

and the obvious one

(7)

For the first condition the potential V23 is modified by the shunting substrateto give

(8)

Introducing the proper relation betweenmeasured current io and applied voltageVo = 2 Ro A io, we find for the desired result

u; (e Re)1/2 (-s -2S)V23 = - -- exp -- exp--BdA A'

(8a)

or writing

u;V23 = -y [exp (-y) - exp (-2y)].

s(8b)

From eq. (8b) it is clear that in a linear geometry at low voltage the values of Reand Ro cannot be obtained separately. However, it will be shown in sec. 2.3 thatboth parameters can be obtained separately when a four-point-probe measure-ment is done with altered probe order. Although this can easily be calculatedalso for this bar-shaped geometry, it will be presented with the more practicalcase of infinite geometry.

2.2. Linear geometry, high voltage

Let us now study the potential distribution when the current io is increasedand the heterotype structure begins to show its diodic character. The zero-biaspoint where the top-layer potential is equal to the substrate potential, thenmoves so that most of the junction is reversely biased and only a small part ofthe junction is-forward-biased with a steeply rising potential in the top layer.Ifthis does not happen, the structure is dominated by leakage and the precedingtheory should be applied.

For a diode-controlled n-on-p" structure instead of eq. (3) the relation

(9)

should be introduced into eqs (1) and (2), which can be combined to

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 285

d2

V + B r: Ro {exp [_!!_ (Vb- V)J- I} = O.dx2 kT

This differential equation can be integrated once to give

(10)

(dV)2 kT { e (Vb - V)- = 2BRoJs- exp----dx e kT

e Vb e V}exp -- +- + (Ro io)2,kT kT

(11)

where dV/dx = - Ro io with Js = 0, and i = io at V = 0 and V = Vo havebeen used as boundary conditions. Because the substrate is not connected to thecircuit, the reverse current should equal the forward current which yields therelation plotted in fig. 3:

3 eVbI- I/ kT=~l- e kT

- /V

- /1

~~ ~~ - eoVb=l ,./ T='+2kT~ IIII I ï I IIIIo

D-l 0·2 0·5· 1 2 5 10 20 50 100~-kT

Fig. 3. The relationship between the floating potential Vb and the applied potential Voo

e (Vb - Vo) e Vb e Voexp = exp --- --.

kT kT kT(12)

Although the linear geometry could actually be used on a heterotype epitaxialstructure by etching two grooves, it is not of primordial interest. Furthermore,the basic assumption of a constant epitaxial-Iayer thickness is certainly notsatisfied here. In eq. (I) the zero-bias thickness of the layer minus the voltage-dependent depletion-layer width should then be introduced instead of d. It hasbeen presented, nevertheless, as far as it can be solved analytically because itclearly illustrates the essentials of the method. In fig. 4 the distribution of volt-age, current and current density has been schematically drawn as a function ofposition along the bar-shaped n-on-p+ layered system. It is clear then that thevoltage V23 is smaller than it is supposed to be and that the voltage drop alongthe reversely biased part of the junction is given by a parabola.

286 P. J. SEVERIN.-------------------------v

. R .V, t'a ,t~3t

a}L -x

b)t[=±Jz L_x

tJs 1----------- .....

c} LI _xIIIIII

Fig. 4. The potential (a), the current (b) and the current density (c) in a bar-shaped n-ea-p-layered system, where the contacts have been taken into account with the appropriate polarityand resistances Rl and R4' and the junction is determined by a diode characteristic.

2.3. Circular geometry, low voltageIn the more realistic cylindrical geometry of a point on a thin layer of infinite

lateral extent the equations corresponding to eqs (1), (2) and (4) are written as

dVer) 1 (2--=---i(r),dr 2nrd

dier)-- = -2n r JzCr),dr

(13)

(14)

and

(15)

with the solution 45)

(16)

where Ko(r/A) is the modified Bessel function of the second kind and thecharacteristic length A = (R; d/(2)1/2. The modified Bessel function of the firstkind is not a solution because it does not satisfy the boundary condition V = Vb

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 287

for r ~ 00. The function Ko(r) diverges for r = 0 and it is difficult to satisfya second boundary condition for a.Integrating eqs (13) and (14) with lz = 0, it is found that the potential is

equal toe io

V = -- (ln r- In 1"1),2nd

where 1"1 is not specified. Since we are only interested in potential differences andespecially in V23, 1"1 can be left undetermined, but eq. (17) determines a.The current io introduced at r = 0 leaves the structure at the boundary at

infinity. When a current of equal magnitude and reverse sign is drawn at adistance 3s, then V23 doubles and no current is drawn at infinity, yielding

e i14V23 = --In 4.

2nd

Applying the same argument to the solution, eq. (16), with finite J, given by eq.(3), the expression

e i14V23 = - {2Ko(y)- 2Ko(2y)}

2nd

is found, where y = sl á = s (elRe dF/2•It is clear that, as in eq. (8b), R; and eld cannot both be found separately from

such single measurement. Therefore, the following procedure is suggested. It isevident that the order of the four current and voltage probes can be changed,and that the resistance values obtained in these modes will be different. Intable I the resistance values for the three possible different permutations arepresented for thin layers with zero-conductivity and with finite-conductivity

TABLE I

Resistance values for three possible modes

mode bulk Ithin layer, R; -+ 00 thin layer, R; finite

eq. (18) eq. (19)

V231 In 4 = 1·38629 2Ko(Y) - 2Ko (2y)R2314 =--

i14V34

0·33 In != 0·28768 Ko(3y)-2Ko(2y)+ Ko(Y)R3412 =--i12V24

,

R2413 =-- 0·67 In 3 = 1·0986 Ko(Y) - ,Ko(3y)i13

(17)

(18)

(19)

288 P. J. SEVERIN

junctions corresponding to eqs (18) and (19), respectively, expressed in unitsel2nd. For comparison purposes the values for bulk material have also been.included, expressed in units el2ns. All otherprobe orders are equal to one ofthese three.It can easily be verified that the relation

(20)

is always valid, hence from the three resistances only one independent ratio,containing y only, can be chosen. From the ratio R23141R2413 or R2314IR341.2'

plotted in figs Sa and b, the value of y can be determined. Inserting this valueinto eq. (19), the sheet resistance eld can be determined from the known value ofR2314• The y-dependent factor in eq. (19), 2Ko(Y) - 2Ko(2y), is plotted in fig. Sc.From y and eld the junction zero-bias barrier resistance R; can be determined,which may be used to calculated the leakage current Js from

kTRe=-·

e i,(21)

The sensitivity of the method can be assessed from the following example.Suppose a S!L thick, I-Q cm n-type layer on a p-type substrate yields R23141

R2413 = I·28 instead of the zero-conductivity value 1·27. Hence the numericalfactor in R2314 equals 1·38 instead of 1'39, with y = 0·03. This corresponds toR; = 104 Q cm? and J, = 2.10-6 A/cm2• Ignoring this effect the sheet resist-ance would have been measured 1% too low.

Tn the next section experiments will be presented which prove that the junctionis diode-controlled and that the altered-probe-order technique can be used inprinciple for obtaining information on the condition of a heterotype structure.However, it will be found that due to the deficiencies of the available four-pointprobe heads no experiment can be done which clearly illustrates the theory. Theexpressions presented in eqs (18) and (19), table I and figs Sa, band c, are allbased on the assumption that all probe spacings are equal and all contactdiameters zero or at least equal. This is not satisfied by several per cent. On theother hand, it has been verified that this non-ideal probe arrangement can veryreproducibly be repositioned. With the appropriate formulas the correctedzero-junction-conductivity values corresponding to eq. (18) and table I can beeasily calculated and the corrected four-point-probe resistivities corresponding :to eq. (19) and table I can be computed.

3. Experimental results

3.1. On the rectifying junction

The hardware involved in the method described are silicon slices and four-

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 289

2

r-,.......<,

t-....l- t---

1

1·2b) 0 02 0·4 0·6 0·8 1·2 M 1'6 1'8

-----<- y2

a)00 0·2 0·4 0'6 0'8 1'2 1-4 1·6 1-8 2

----y

1·5

~

~ V

LV

V-:/'

L_V

1·6

1·4

1-3

1-6R23M

~~dJ 1·4

11,2

1·0

0·4

<,~<,I'-."",

r-.<,r-,

.............l"-r-

0'8

0·6

0·2

1'2 1-4 1'6 1·8 2-yc)

Fig. 5. The ratios R2314/R3214 (a) or R2314/R2413 (b) as a function of y are used to calculatey. From the value of y the resistance R2314 can be read in units e/27rd, and compared withthe measured value (c).

290 P. J. SEVERIN

point-probe heads. Both are imperfect to an extent which makes it difficult toindependently test the theory or one of the two components. The probe headsused are Dumas (80 & 200) and Fell manufactured"). The results should beindependent of the instrument used; if they are not, one or neither of them canbe used to prove the theory with sufficient accuracy. The material should be ofsufficiently uniform resistivity and zero-bias junction resistance so that theeffect predicted by the theory does not disappear due to lack ofprecision.As a first experiment to qualitatively test the model on which the theory is

based, the potentialof each ofthe two centre probes, 2 and 3, has been measuredwith respect to the substrate potential on a large number of n-type epitaxiallygrown layers on p-type substrates with a Dumas probe. They all show quali-tatively the same dependence on the current i14, shown in fig. 6. For one probe,3 in fig. 6, the potential Vb3 first increases with increasing i14, reaches a maxi-mum, decreases, changes sign and then V3b increases further. The maximum inVb3

is found generally at about 60 mV at current levels depending on the sheetresistance concerned. The potential Vb3 vanishes when the zero-biased point ofthejunction isjust below probe 3. The potential difference V23 is generally quitewell proportional to i14, and V2b bends such that this holds.

Normally four-point-probe measurements are done at a rather-high-voltagelevel corresponding to about 1 mA. As explained in the preceding section, theepitaxiallayer is then better isolated from the substrate, but on the other handthe depletion-layer width is increased and position-dependent. Several series of

.1·6I"(mA)

tM1·2IV3~L--"

Vr-r<,

~ ~ ~ ~V

.'\1// ~

1/ V ;144./ (mA)] ~~ t..- ~b I

J.;V Vt ~

i"o' ~ I- fti;;;_-t..- I--"? I-- ~I)//_/V .-

V I

~V I ,DO 1400 600 800 T~V(mV) I

1·0

0·8

0·6

0'4

0-2 1000

20 40 60 80 100 120 140 160 180 200-V(mV)

Fig. 6. Typical results of the dependence of V2b and V3b on the current i14 with an Il-type-Siepitaxially grown layer on a p-type substrate (H260) : d = 11·3 fLm,(lId = 650 n.

*) The Dumas Instrument Company, Elmat Corporation, Mountain View, California;A & M Fell; Lambeth High Street, London.

FbuR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 291

heterotype structures grown as one batch have been four-point-probe measuredin the centre of each slice and the results of two series are plotted in fig. 7. Thevalues of R2314 are for all slices systematically higher at millivolt level, corre-sponding to a few fLA, than at the usual high-voltage level corresponding t?0·45 mA. This effect typically amounts to about 2% for about 1-Q cm, a fewmicrons thick n-type layers on p-type substrates. In view of the theory outlinedin the preceding section it is advocated that four-point-probe measurements bedone at millivolt level and that the increased shunting effect eventually occurringis investigated with the altered-probe-order technique. The high-voltage situa-tion is not easily amenable to theoretical analysis and not reproducible due tothe dependence on the current level. It is worth noting that this type of depend-ence on the current level does not occur always, even if the junction is diode-controlled, as fig. 6 shows. The sum rule eq. (20) has been found not to be validwhen the system behaves non-linearly. This is a safe criterion.

In the model assumed above the voltage drops across the two currentprobe-silicon contacts, shown in fig. 4 as ioRt and ioR4' have been neglected.With a Dumas four-point probe the probe-silicon contact is spreading resist-ance-controlled at low voltage 46). The actual value depends on age, appliedpressure and many other parameters, but a value equal to about e/4A*,where A* ~ 10 fLm, is never exceeded. Twice this value should be comparedto the sheet-resistance contribution in the current circuit which is equal toabout e/d. With other probe materials the zero-bias barrier resistance can alsocontribute to the contact resistance, which in that case is not exactly propor-tional to e. In general the probe-contact resistances are at most of the sameorder of magnitude as the sheet resistance, so that the millivolt condition ishardly relaxed due to this departure from the assumed model.

...--------_,/./-/_--------,i i

: Ix • •

x • •

. . . . . . . .1·297±0·006 1'292± 0·006

'---'--L.-'-_'_-'--'---L-.....y/ ",'Cl./ -'--'---'--L....J.--L_J_....J

R2/'72/B i / R21,72/F

Fig. 7. Two series of eight slices four-point-probe (Dumas) measured in the centre at millivoltlevel (upper dots) and at the usual current level (crosses). The epitaxiallayers are /I-type 15 [Lmthick and 1'2-n cm. The ratio R2314/R2413 is shown to be almost constant (lower dots).

292 P. J. SEVERIN

3.2. On the altered probe order

In order to eliminate the effects of individual slices, the value of R2314 and ofR2314/R2413 are measured with different probe heads on the same position,usually the centre, of a number of slices grown as one batch. In all cases studiedit has been found that the measured values obtained with one type of probehead can be connected to form a curve at about constant distance from the curvecorresponding to another type ofprobe head. The deviation from such constantdistance is smaller than the distance between the curves. The reproducibility isshown to be even better when the measurement series is repeated at about thesame points. In other words, the precision is much better than the accuracy.Typical examples of such behaviour are shown in figs 7 and 8. The Fell probeyields a value of R2314 about 10% lower than the Dumas 80 &200 probes whichgive values that are about equal. Furthermore, the Fell probe suggests that acorrection for substrate shunting is needed to R2314, whereas the Dumas probesyie!d values for' R2314/R2413 sometimes even below the theoretical limit forzero-junction conductivity. This has been checked also on substrate sliceswhichare thin enough to be described by the preceding theory, and which should showthe correct zero-junction-conductivity values.The diameters Al of the prints of the probes and their centre-to-centre

spacings sI) have been measured. Because most probes show sliding, accuratedetermination is fairly difficult. Reproducible departures from constant spacingup to a few per cent have been found with different probes, whereas the contactdiameter also depends upon the age. It can easily be calculated from eq. (17)

R2314(.aJ

t. . :- 200. . . -- 180-- 160-- I~O

- 120

R2314(I1J

t18001-

. .1·2

1·0

1600f- ! 0

1~00 l- .l-

• 0

1200 I-I- •

1000 I- 0 •.f-

800

H128-2 PI

Fig. 8. Two batches of eight slices four-point-probe measured with a Dumas 80 (dots) and200 (circles) and with a Fell (crosses) four-point probe. The ratios R2314/R2413 are similarlyplotted and shown to be probe-type-dependent, but constant.

FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 293

that when the different spacings and contact diameters are taken into accountin the thin layer, the R; -+ 00 data of table I turn out to be special cases of themore general expressions

and(SI3- Al) (S24- A2)

R3412,..., ln---------(S14 - Al) (S23 - A2)

When the measured data on Al and slj are taken into account, with eq. (22) thecorrect value for the ratio is found for the particular probe head.In figs 7 and 8 the ratio R2314/R2413 has also been presented, as measured

with the different types of probe. It is evident from these figures that with amechanically stable probe the ratio can be very carefully measured, up to ± 1%.The sheet resistance is then accurately known ± 0·5%.

3.3. On the uniformity of R; and Ro

As an elementary check to test the uniformity in Ro and R; the potentials Vb2and Vb3 are measured at the low-voltage level referred to in the precedingsections. Ideally they should be equal, but it has been found that their ratio onmany slices scatters rather statistically and on other slices shows a certainsystematic distribution. The actual value of Vb2/Vb3 does not change when thecurrent is reversed, as is to be expected in a linear system. However, when theslice is turned over 180Q and the probe is positioned carefully on the same spot,then the point of zero bias is found near the other end of the probe. In otherwords, the ratio Vb2/Vb3 is determined by the local non-uniformity in Ro and/orRc, be it of an irregular, statistically distributed or a systematic nature due to thegrowth process. When R; is high enough, non-uniformity in R; is not manifestedin R2314, nor in R2314/R2413' Any non-uniformity in Ro influences the resultsmeasured in R2314 and in R2413• Ifthe non-uniformity is substantial on a scalecomparable to the integration length of the four-point probe, neither the twofour-point-probe resistances, not their ratio can be ofuse. Ifthe non-uniformityis small correct four-point-probe measurements may be possible. This condi-tion can be detected by a technique which can be considered as a combina-tion of the spreading-resistance and the four-point-probe technique and whichwill be discussed in a subsequent paper 47).

3.4. On high-frequency signals

Measuring heterotype epitaxial structures at current levels commonly used

(22)

294 P. J. SEVERIN

for four-point-probe measurements, e.g. 0·45 mA with a I-Q cm, 5 [L thickn-on-p-type layer, sometimes discrepancies were found which could not beexplained at all. It turned out that in the four-point-probe circuit quite largea.c. signals can be generated of complex waveform with periods in the range10-5 < 7: < 10-7 s. The frequency f= 7:-1 and the amplitude of the a.c.signal increase monotonically with the d.c. current i14 from a certain valueonwards. No 11-9n-n+ and not all n-on-p or p-on-n slices have been found to beable to generate h.f. signals. The signal is harmonic when the system is on theverge of oscillation, but changes to a more triangular shape with increasingfrequency. On a 2X 3-mm2 rectangle, isolated by etched grooves from the restofthe slice, the signal showed a more harmonic waveform. The r.f. voltage canbe a substantial fraction of the applied d.c. potential, e.g. 20%. Therefore, it isevident that a four-point-probe measurement can be severely hindered by thiseffect. It is fairly reproducible and uniform all over a slice. A number of typicaldata are presented in fig. 9.It does not occur at the low-voltage and corresponding current levels where a

four-point-probe measurement should be made, as discussed in this paper.Therefore, the cause of this effect will not be investigated. It has been verified bymeasuring simultaneously the different probe potentials that the effect is not dueto a moving domain. The experiments were done in a shielded cage.

4·0V(V)

13,5

3-D

2-D

/V

./V' -

1/ VJ

II1/ -

I~~ ~

-.hvx

~-

IS

(}S

((MHz)

2-Ot

2·5 1·5

1·0'1·5

1·0

().s

oo 2 4 6 8 10 12_i" (mA)

Fig. 9. High-frequency effects occurring with a normal four-point-probe measurement on a5·3 {Lmthick, 0'7-0 cmp-type Si layer on /I-type substrate. The dependence on the current i14of the following parameters is shown: V23 (triangles), 2V23 (squares) referring to the left-hand axis and the frequency f (crosses) referring to the right-hand axis.

FOUR-POINT-PROBE MEASUREMENTS ON 'Si EPITAXIAL LAYERS 295

4. Discussion

In order to discuss the precision and accuracy of the method it is worthwhileconsidering the validity of the assumptions deliberately or tacitly made. It wasdeliberately assumed that the layer thickness is uniform and small with respectto the probe spacing s. The second condition is reasonably well satisfied withsubstrates and very well with epitaxial layers. The three probe spacings shouldbe equal and reproducible. That they do riot satisfy the latter requirement hasbeen shown to affect the precision, that they are not equal may effect the accuracy.It has been assumed that the contacts are mathematical points of infinite con-ductivity, whereas actually 2A Rj 20 Il. for Dumas and about 30 Il. for Fell probesand the contact resistance is by no means negligible. The detailed nature of thesteel probe contact to silicon for spreading-resistance measurements hasbeen dealt with by Severin 46) and its use on heterotype epitaxiallayers will beelaborated elsewhere 47). It can easily be verified that to a first approximationthe system is determined by sheet resistance between the outer edge of thecurrent-probe contact and the centre ofthe voltage-probe contact. The differentspacings and the finite contact radii yield values of the ratio R2314/R2413 onisolated thin layers typical of a particular probe. This characteristic ratio shouldbe checked regularly. This value should be taken as the unit value in figs Sa, band c, which cannot be used for accurate measurements, as explained in sec. 2.3.It has explicitly been stated that the system refers only to systems of infinite

extent in the horizontal plane. For finite systems the solution to eq. (15) .shouldalso include modified Bessel functions of the first kind with specified boundaryconditions. Therefore some disagreement with the theory can be expected atabout 2 mm from the edge of a slice.

5. Summary and conclusions

When the four-point-probe resistance R2314 of a heterotype epitaxial struc-ture is not independent ofthe current level corresponding to low or high voltage,the layer is shunted by a substrate with a non-linear junction characteristic ofrelevant conductance. Whether the junction is diode- or resistance-controlled,can be determined by measuring Vb2or Vb3 vs i14• If the junction is found to bea resistance, the value can be determined by the method of altered probe orderat any current level. When the junction is diode-controlled, the measured four-point-probe resistance is determined by sheet resistance, junction conductanceand unknown non-uniform, current-dependent depletion-layer width.

Although, of course, at high-voltage level the layer and substrate are betterisolated than at low-voltage level, these depletion-layer properties precludereliable measurements under the former conditions. It is advocated to do themeasurement under the defined conditions which exist at low-voltage level andto allow for the increased junction leakage with the formalism described. It

296 P. J. SEVERIN

consists essentially of measuring in addition to the normal four-point-proberesistance R2314 also the ratio R2314/R2413' This ratio turns out to be a goodcharacteristic parameter for the probe quality as far as differences in probespacings and contact diameters are concerned.

AcknowledgementThe assistance of Mr. G. Vermeulen is gratefully acknowledged.

Eindhoven, April 1971

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FOUR-POINT-PROBE MEASUREMENTS ON Si EPITAXIAL LAYERS 297

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