'. ' 220 HILlS TECHNICAL REVIEW VOLUME 20 A M,ETHOD OF MEASURING THE RESISTIVITY AND HALL' COEFFICIENT ON LAMELLAE OF ARBITRARY SHAE 621.317.331:538.632.083 Resistivity and Hall-coefficient measurements at different temeratures lay an imortant art' in research on semiconductors, such as germanium and silicon 1), for it is from these quantities that the mobility and concentration of the charge carriers are found. Such measurements are commonly. carried out with a test bar as illustrated infig. 1. The resistivity is, found directly from the otential difference and the distance between the contacts 0 and, the current i and the dimensions of the bar. To determine the Hall coefficient the bar is subjected to a magnetic field B alied at right angles to the direction of the current and to the line connecting the diametrically oosite contacts 0 and Q. From the otential difference thus roduced between these latter contacts the Hall coefficient is derived. (The relation between the Hall coefficient and the change in electric otential distribution due to a magnetic field will be exlained resently.) o samle is that it is rather difficult to make, having to he cut out of the brittle semiconductor material with an ultrasonic tool. There is therefore a considerable risk of breakage, articularly when the arms are made narrow. Fig. 2. The bridge-tye samle, which is rovided with relatively large contact faces to reduce contact resistances. This form is of secial imortance in measurements at low temeratures. In the following we shall describe a method of erforming resistivity and Hall-coefficient measurements on lamellae of arb.itrary shae 2). The lamella must not, however, contain any (geometrical) holes. Q R 95924 Fig. 1. Classical form of samle used for resistivity and Hallcoefficient measurements. The test bar is rovided with current contacts M and N and voltage contacts 0,, Q and R. The. fourth voltage contact R serves for check measurements. In measurements erformed at low temeratures - e.g. in liquid nitrogen - oint contacts ossess resistances of the order of megohms, in consequence of which the voltages cannot be determined with sufficient accuracy. In such cases "bridge-shaed" samles are used as illustrated infig. 2. The voltage and current contacts here have a relatively large surface area, and, hence the contact resistances are low. The methóds referred to are based on the fact that the geometry of the samle ensures a simle attern of virtually arallel current stream-lines.. Formulae have been devised to correct for the deviation from arallelism in fig. 2, caused by the finite width of the arms. A drawback of the bridge-shaed New method of measuring resistivity We take a flat lamella, comletely free of holes, and rovide it with four small contacts, M, N, 0 and, at arbitrary laces on the erihery (fig. 3). We aly a current i MN to contact M and take it off at contact N. We measure the otential difference V - V o and define: Analogously we define: The new method of measurement is based on the theorem that between RMN,O and RNO,M there exists the simle relation: ex (- :ned RMN,O) + ex (- :n: RNO,M) = 1, (1) where d is the thickness of the lamella and e the 1) See e.g. C. Kittel, Introduetion to solid state hysics, 2nd edition, Wile,y, New York 1956, Chater 13,. 347 et seq. 2) L. J. van der auw, A method of measuring secific resistivity and Hall effect of discs of arbitrary shae, hilis Res. Rets. 13, 1-9, 1958 (No. I).
1958/59, No. 8 RESISTIVITY AND HALL COEFFICIENT _ 221 resistivity ofthe material. If d and the "resistances" RMN,O and RNO,M are known, then (1) yields an equation in which e is the only unknown quantity. Fig. 3. A flat lamella of arbitrary shae, with four contacts M, N, 0 and on the erihery, as used in the new method of measuring resistivity. The Hall coefficient can also be measured on a samle of this kind. The situation is articularly straightforward if the samle ossesses a line of symmetry. In that case, M and 0 are laced on the line of symmetry, while Nand are disosed symmetrically with resect to this line (fig. 4). Then: which may be seen as follows. From the recirocity theorem for assive fouroles, we have quite generally that RNO,M = RM,NO (interchange of current and voltage contacts), and it follows from the symmetry that RM,NO = RMN,O' Hence we arrive at (2); e can then easily be found from (1): nd e = ln2 RMN,O' (3) (2) In the general case it is not ossible to exress e exlicitly in known functions. The solution can, however, be written in the form _ nd RMN,O + RNO,M f e - ln2 2 ' wher'ef is' a factor which is a function only of the ratio RMN,O/RNO,M' as lotted in fig. 5. Thus, to determine e, we first calculate RMN,O/RNO,M' read from fig. 5 the corresonding value of f and then find e from (4). hotograhs of samles as used for the old and for the new method' are shown in fig. 6. The comlete roof of the theorem underlying the measurement of e is given in the aer quoted in footnote 2). The roof consists of two arts. First of all, relation (1) is develoed for a secial case, the case of a lamella in the form of an infinite half-lane, rovided with four contacts at the 0,8 t jo,6 0,4 0,2 -- ---... <, r-,r-; ;-... -r-- T 2 5 70 2 5 10 2 2 5 T0 3 _ RMN,O (4) RNO,M.5928 Fig. 5. Between the factor f in formula (4) and the ratio RMN,O/RNO,M there exists the relation: h S (RMN,O/RNO,M) -1 In2l 1 In 2 cos? (RMN,OjRNO,M) + 1 f ~= "2- ex7' which is reresented here grahically. In this case a single measurement suffices. erihery. It is then shown that the relation must also aly to a lamella of any shae. This is done by means of conformal maing of the arbitrarily shaed late on the infinite half-lane with the aid of comlex functions. N 95927 Fig. 4. The resistivity measurement is simlified if the samle has a line of symmetry. If two of the contacts are situated on the line of symmetry and the two others are symmetrically situated with resect to this line, one measurement is sufficient to give t~e required resistivity. We shall consider the first art of the roof in more detail, since it reveals the origin of the exonential functions in (1). We first consider a lamella which extends to infinity in all directions. To a' oint M we aly a current 2i, which flows away from M with radial symmetry into infinity. Let dagain be the thickness of the lamella and e the resistivity. Then at a distance r from M the current density is J = 'i.ij2nrd. The field-strength E is radially oriented and according to the generalized form of Ohm's Iaw has the value: E = ej = ei/nrd.
222 HILlS TECHNICAL REVIEW VOLUME 20 a b c d lcm 959'12 Fig. 6. Some samles of silicon used for resistrvity and Hall-coefficient measurements. Samles a and b corresond resectively to figs. 1 and 2. Measurements on samles c and d are ossible only by the new method. The incisions in samle d serve to reduce tbe error caused by the contacts not being infinitely small. The otential difference between two oints 0 and lying on a straight line with M (fig. 7a.) is: a V-Va=(Edr=l.!i (dr=_l.!i lna+b+c. a. =«. r rul a + b Since no current flows in the direction erendicular to the line through M, 0 and, the result obtained remains valid if we omit the art of the lamella at one side of this line - yielding a half-lane - and if at the same time we halve the current (fig. 7b). Next we consider the case of c) in fig. 7, where a current i now flows out from a oint N, that again lies on the same straight We now find: Suerosition line with O, viz. on the edge of the infinite half-lane. I.! i b+ c V - Va = + nd In -b-. of the cases b) and c) in fig. 7 yields the case d), the current i being introduced at M and taken off at N. The value now assumed by V - Va is found by adding together the two revious results. After dividing by i we then find: Formula Next (1) can then be written: we write: Xl x:! e e+e Q=I. (6 ) Xl =.}{(Xl + X 2 ) + (Xl - x2)} and X 2 = t{(x 1 + x 2 ) - (xl -x 2 )}, whereby (6) takes the form: This is the same Xl +x'! ~ X1-:\:2 + X_"_-_:\:Z e- ~ (e 20 + e 2e) = 1. as: The exonent of e in (7) is now written as -(In 2)/1, i.e. we ut: Formula (7) then becomes: x I +x 2 1n2 21.! =1' (7) (8 ) or Similarly R g_ I (a + b) (b + c) MN,a-nd n (a+b+c)b ' (a + b + c)b ( rul \ (a + b) (b + c) = ex -(i RMN,a/. we find: ac (nel \ (a + b) (b + c) = ex -(i RNa,M/' g 2i~I<>.- Ivf 0 ~a~+~b~---_.1j..-----,,-c_-_j Addition of the last two equations yields (1). We shall now exlain how formula (4) follows from (1). For simlification we ut: n. d RMN,a = Xl' ) st d RNa,M = X 2 ) (S) Fig. 7. Illustrating the derivation of formula (1).
1958/59, No. 8 RESISTIVITY AND HALL COEFFICIENT 223 This exression reresents a relation between f and Xr/x 2, and hence also between f and RMN,O/RNO,M (see 5). The relation is shown grahically in fig. 5. By re-writing (8) to give (! and substituting for Xl and X2 from (5), we find formulà (4). Method of measuring the Hall coefficient The Hall coefficient, too, can bè measured on an arbitrary lamella as in fig. 3. We then aly the current to one of the contacts, say M, and take it off at the contact following the succeeding one, i.e. in our case at O. We measure RMo,N' after which we set u an uniform magnetic induction B at right angles to the surface of the lamella. This changes RMo,N by an amount LlRMo,N' We shall now denote the Hall coefficient RH and show that it is given by: EH is roortional to J and to B; the roortionality constant (= ljnq) is called the Hall coefficient RH' Since q. is known, one can calculate from RH the concentration n of the charge -carriers, The fact that the current stream-lines are not affected by the magnetic field imlies that after alication of the magnetic field, the electric field is no longer' in the same direction as the current stream-lines, but has acquired a transverse comonent Et' which exactly comensates the (9) rovided that: a) the contacts are sufficiently small, b) the contacts are on the erihery, c) the lamella is of uniform thickness and free of holes. The validity of formula (9) deends on the distribution of current stream-lines not changing when the magnetic field is alied. With samles of the classical shae of figs. 1 and 2, where the current stream-lines are always arallel to the edges of the samle, there is evidently no change. From the roerties of the vector field reresenting the current density it follows that the same also alies to lamellae of arbitrary shae, rovided the above conditions are satisfied 3). Under the magnetic induction B,,the charge carriers, with charge q, are subjected to a force erendicular to the stream-lines and erendicular to the lines of magnetic induction. The magnitude of this force is F = qvb, where v is the velocity ofthe charge carriers. Between v, the concentration n of the charge carriers and the current density J there exists the relation v = Jjnq. Dividing the force exerted on the charge carriers by their charge q, we see that the effect of the magnetic field is equivalent to an aarent electric field EH' the Hall electric field, for which we can write 4): Fig. 8. The resultant of the electrical field-strength E and the Hall field-strength EH lies in the direction of the current density J. Resolving E in directions erendicular and arallel to J therefore yields a erendicular comonent Et which in magnitude is equal to EH. aarent Hall electric field EH (fig.8). The change' LI(V - VN) in the otential difference between and N is therefore given by integrating Et from?ver a ath orthogonal to the current stream-lines to N'. across the lamella (fig.9), and thence along the erihery -- i.e. along a stream-line - from N' to N. The last ortion of the ath makes no contribution to the integral; hence N' N' LI(V-V N ) =f EHds=RHB f Jds=RHB i~o, where d is again the thickness of the lamella. This exression leads directly to (9). 1 E H = -JB. nq 3) The roof of this statement is also indicated in the aer quoted under 2)...4). 'I'his.formula is not entirely exact because, aart from their. ordered motion with velocity v, the electrons also move randomly owing to thermal agitation. More recise calculation shows, however, that the formula given here is a good aroximation. Fig. 9. To calculate by how much the otential difference between oints. and N changes when a magnetic field is alied at right-angles to the lane of the samle, the transverse electricfieldetroduced by the magnetic fieldis integrated along the ath s which runs from, orthogonal to the current. stream-lines, to N' and thence along the erihery from N' ton..
HILlS TECHNICAL REVIEW VOLUME 20 Estimation of errors In the foregoing we have assumed the contacts to be "sufficiently" small and to be situated on the erihery. To gain an idea of the error made in the calculations when these conditions are not exactly satisfied, we have worked out the error for three cases. For simlicity we consider a circular disc of diameter D with the contacts mutually 90 aart. We assume further that only one of the contacts is not ideal. The three cases are reresented in the adjoining table, together with the formulae for the relative errors in the resistivity and the Hall coefficient. The cases are: a) One of the contacts has. a length I along the erihery. b) One of the contacts has a length I erendicular, to the erihery, c) One of the contacts, although a oint, is situated at a distance I from the erihery. In ractice, none of the contacts will be ideal. To the first aroximation the total error is then equal to the sum of the errors er contact. The value of the' method described here lies in the fact that, in many cases, the material under investigation is already available in the form of small lamellae (e.g. thin discs sawn from a crystal rod); these samles now need no further rearation and can therefore be used for other uroses after the measurement. If very small contacts are undesirable, having regard to contact resistances in measurements at low temerature, use can be made of "clover-leaf" Table. The relative errors.dele and.drhirh in the calculated values of the resistivity and the Hall coefficient for a circular disc of diameter D on which one contact is non-ideal, in the senses indicated in the sketches. s Qo _12 M, D.dele N - ARH/RH -21 ;:::::760 2 ln'2 ""-- "-' 3(20!~I _12 -~I ""-- "" ~021n2 ~---;(2ï) D M~ jo 1\ ~ MÇ)o _1 N 2-21 "--- ~ 2{hn2 "- :rto samles (see fig. 6d), the incisions in which substantially reduce the error due to the finite dimensions of the contacts. The clover-leaf samle thus relaces the bridge-tye samle (fig. 6b) which is used for the same urose in the classical method. The clover-leaf samle is easier to make than the bridge-tye samle and is also less suscetible to breakage. L. J. van der AUW.