ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE

Transcription

1 R 275 Philips Res. Rep. 10, , 1955 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE by J. van den BOOMGAARD Summary The mathematical description of the zone-melting process, as given by Pfann and by Reiss only concerns the cases of ingots consisting of an element or a nondecomposing compound with non-volatile solutes. In this paper the theory is extended to elements containing volatile solutes. As will be shown, it is possible to bring a volatile impurity element homogeneously into an ingot by means of zone melting under a constant vapour pressure of that element. Résumé La description mathématique du procédé de fusion par zone, donnée par Pfann et par Reiss, ne concerne que le cas de lingots se composant d'un ëlëment, ou d'une combinaison ne se dëcomposant pas, avec des substances dissoutes non-volatiles. Dans eet article, la théorie est étendue à des ëlëments contenant des substances dissoutes volatiles. Comme on verra, il est possible d'introduire une impureté volatile de manière homogène dans un!ingot au moyen de la fusion par zone sous une pres sion de vapeur constante de eet ëlëment. Zusammenfassung Die mathematische Beschreibung des Zonenschmelzprozesses, wie sie von Pfann und von Reiss gegeben wird, behandelt nur den Fall von GuBblöcken, die sich aus einem Element oder aus einer nicht zerfallenden Verbindung mit nicht flüchtigen Bestandteilen zusammensetzen. In diesem Artikel wird die Theorie auf Elemente ausgedehnt, die flüchtige Bestandteile enthalten. Es wird gezeigt, dall es möglich ist, ein flüchtiges Fremdelement gleichmällig in einen GuBblock zu bringen, und zwar durch Zonenschmelzen unter konstantem Dampfdruck des betreffenden Elements. 1. Introduetion In the last few years, zone-melting processes have often been used in order to get pure materials or materials with a homogeneous distribution of certain foreign atoms. In calculating the concentration as a function of 'place, Pfann 1) introduced a number of simplifying assumptions, which, while not entirely realizable in practice, nevertheless provided a suitable point of departure for more refined treatments. Reiss 3) gave some mathematical methods for zone-melting processes. These authors, however, only considered ingots with non-volatile solutes. In this paper the zone-levelling process of a volatile solute in an element under a constant vapour pressure of that solute will he considered. With some simplifying assumptions mathematical expressions will be derived for this case.

2 320 J. van den BOOMGAARD 2. Experimental conditions When a solid which tends to decompose is heated without any precaution being taken, the substance will decompose with the formation of a vapour consisting mainly of the volatile component. Decomposition can be prevented by maintaining a constant pressure of that component over the solid. Various methods of doing so have been described elsewhere 4). The same holds for liquids which tend to decompose and for liquids and solids containing what is under these circumstances a volatile solute. In the present paper we refer mainly to the methods mentioned above, using a closed evacuated vessel, in which the solid and the pure volatile solute are heated in a temperature gradient as shown in fig. 1. Ec=lHFcoiZ C coo/ring =:!I=~~2.r.!LI... L.:-:-:-=-=-=-=-=-~~~A vja zz zz zz z1zz zz zz zzz?z 21 ) -0 c=le ~--~ ~ Fig. 1. Schematic drawing of an apparatus for zone-melting processes in a controlled atmosphere. At A the solid is present, at B the pure volatile solute. T 2 is chosen below the melting point of the solid but above Tl so that Tl (measuredby a thermocouple in tube C) determines the vapour pressure of B present in the tube. Zone melting is achieved by increasing the temperature at a particular section sufficiently for melting to take' place and when this high-temperature zone moves, the molten zone moves along with it. 3. Zone levelling under a constant vapour pressure of the impurity element If a molten zone of a definite volume is moved with a linear velocity v through a rod of A under a constant pressure of the element B (see fig. 2)

3 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 321 there are three effects, owing to which the composition of the melt and thus the composition of the solid in equilibrium with that melt may change, viz. (1) reaction between liquid and vapour; (2) segregation at one end of the molten zone; (3) dissolving of solute at the other end of the molten zone. o-x Cx solid x -V Co solid (L~l), L 1}S537 Fig. 2. Solidification hy zone melting (schematic). The composition of the liquid will change as long as the number of B atoms entering the zone differs from the number leaving it. After a certain time, however, these numbers will practically be equal. Then a stationary concentration of B in A is reached. The value of this concentratien will depend on the length of the molten zone, its volume, its velocity and the applied pressure of B. The other quantities which will have an influence on that value are given by the properties of the chosen elements A and ;S such as the dissolving rate of B in liquid A and the distrihution equilibrium of B between liquid and solid. 3.1 Assumptions In order to calculate the concentration of B as a function of place in the rod of A the following assumptions have been made: (a) The diffusion in the liquid is extremely rapid, i.e., the solute is always homogeneously distributed in the liquid. (b) The diffusion in the solid is slow, i.e., the composition of the solid does not change after having grown from the liquid. This holds for reactions both between liquid and solid and between gas and solid. (c) A possible equilibrium in the gas phase, such as Bn ~ nb is reached at once. (d) B dissolves atomically in A. (e) The distribution equilibrium between liquid and solid is adjusted at once. (f) The distribution coefficient for the equilibrium between liquid and solid is a constant, and independent of the composition. 3.2 The zone-levelling process If a rod of the element A with length L and a constant cross-section of 1 cm 2 has a "homogeneous" *) concentration Co of B, and if a molten zone *) "Homogeneous" is meant here in the sense that the concentration may vary throughout a given cross-section, hut that its average over the cross-section is invariant with x.

4 322 J. van den BOOMGAARD with constant length l is passed through it at a constant speed v under a constant pressure of the element B, then the concentration of B in the molten zone, called rx, can be calculated as a function of x with the aid of the three effects mentioned above. Ad 1 : The reaction rate of the liquid with the gas phase can be described by (1) in which k; is a reaction constant and req is the concentration of B in the liquid in equilibrium with' the gas phase. Because the reaction rate is proportional to the number of atoms entering the melt in a time dt and inversely proportional to the volume V of the melt, and because the number of atoms entering the melt from the gas phase is proportional to the surface area S of the melt through which the reaction can take place, V <5= - (2) Ṡ As the zone moves with the constant velocity v, and vt= x dx dt=-. v (3) With the aid of (1) and (3) the first change in rx is (4) Ad 2: The quantity distance dx, will be of the solute frozen out, after advancing the zone a (5) in which k is the distribution coefficient under equilibrium conditions. The change in the concentration in the liquid will he Ad 3: The quantity of the solute entering the zone by melting a part dx of the charge is (7) (6)

5 Jt V. P HI1IPS' GLOEILAMPENFABRIEKEN ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 323 This will change the concentration in the liquid by an amount Co dar.x = - l The net change in rx is therefore dx. (8) S (k kt) kt 1 2 ar, = (- l + öv r,+ öv req +l CoS dx. (9) The concentration in the solid in equilibrium with the liquid is Cx= krx in which C x is the concentration at any x between x = 0 With the aid of (9) and (10) we obtain ac, (k kt)». k Cx = - Ceq +- Co' dx l Ov ov l (10) and x = L-l. (11) The solution of (11) is - (!:-z» ktlceq + kov Co ( - (!: + ~)") C = C' e I 6u e I 6u x. 1 krl + kov in which Ci is the initial concentration for x = 0 (see fig. 3). For an infinite rod a stationary state is reached in which Cx has a constant value: (12) C _ ktl Ceq köv Co st - ktl + köv (13) The value of Ci depends on the way in which the experiment is carried out. Ifthe first zone length l (for x = 0) is molten and has started to move at exactly the same moment when the reaction with the gas phase begins, then Ci = kco' (L-VL Fig. 3. The concentratien of the volatile solute as a function of place after passing one zone.

6 324 J. van don BOOlllGAARD In practice, however, this is not well possible. Ci has a different value which may, however, also be calculated to a good approximation for a procedure in which after the first zone length Z is molten a time -c is waited before moving it. This time -c must be long in proportion to the time nec1!ssary to melt the first zone length Z. With the aid of (1) and (10) it is easy to derive which together with (12) leads to k, Ci = Ceq - (Ceq - kco) e- {;T, (14) kr (k kr) ~ --T~ _ - +- x Cx = ~Ceq - (Ceq - kco) e e S e 1 óo Interpretation of formula (15) ~vk Co + krz Ceq ( + l-e ó. ~vk + krz _ (~+ ~)x) (15) Depending on the values of the various constants eq. (15) may be simplified in various ways. (a) For k; eq. (15) changes into c; = Ceq. (16) Now the equilibrium between gas and liquid is reached at once, and the rod becomes homogeneous with a concentration Ceq. (fj) If there is no, or only an e:xtremely slow, reaction between the gas and the liquid (kr = 0) eq. (15) changes into c,.s., -=l-(l-k)e 1 (17) Co This is the same formula as derived by Read, and used by Pfann 1) for the traditional zone-levelling process. (y) For kr/~v-:?>k/z eq. (15) changes into _!:.!.T _~'" ~vkco + krzceq ( _~"') Cx= SCeq- (Ceq-kCo) e s ~ e óo + 1- e óo (18) ( S ~vk + krz For Co smaller or of the same order as Ceq this formula further simplifies to _~ (T+~) Cx= Ceq- (Ceq-kCo) e Ó v. (19) Thus Cx depends only on the sum of the times xl», necessary to reach the point x and r, during which the zone has been in the point x = O. In this case the samc concentration will therefore always be found at a given

7 ZONE MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 325 point x if different velocities v are used and the waiting time 't is varied in such a way as to keep 'i + x]» constant.' 3.4 The stationary concentration From (12) it was seen that the concentration in the solid tends to reach a certain stationary value Cst, which differs from the equilibrium coneentration Ccq. Since we are interested in a rod of A with a homogeneous concentration of B, this concentration is very important. The value of Cst is - Co + - Ccq ~vk Co + krl Ccq 1 ~v Cst = = ~vk + krl k kr l + ~v k k r (13) The value of Cst for given Co, Ceq (PB)' k and k r depends on the values of v and l: If kr/~v and kjl do not differ very much we can choose v and 1 and reach a certain Cst. With other combinations of v and 1 a different Cst is reached. If by varying the experimental conditions in a suitable way three different values of Cst are reached at the same pressure PB' the quantities k; k; and Ceq can be calculated provided Co is a known quantity. For kr/~v,:?>k/l a value of Cst is reached which is practically equal to Ceq. For k/l,:?>kr/~v the value of Cst is practically equal to Co. If we call kr/~v = a kil, the course of Cst as a function of a is given in fig. 4. To give a numerical example: 1 = 1 cm, v = cm/sec and ~:::::Ilcm. For equal values of k and k" kr/~v :::::1500kil. In that case Cst is practically equal to Ceq. If we put k = 10-2 (which is a normal value for the distribution coefficient) k r has to be cm/sec to obtain equal contributions of Co and Ceq in Cst, viz. Cst = t(ceq + Co), Ceq :.;: ~~----~-~ _11: Fig. 4. Cst as a function of a = krl/kov.

8 326 J. van den BOOMGAARD If the quantities k, k" Ceq are known, and a value for 1 and v has been chosen, then it is possible to calculate which value has to be given to -r to obtain a rod with a homogeneous concentration Cst between x = 0 and x = L...,--l. In this case Ci = Cst and thus -rst = ~ In [(1 + ktl) (C eq - kc o )] k; övk Ceq - Co (20) For Co = 0 or Co ~ Ceq, we obtain -rst = -ö In ( 1+ - ktl),. kt övk (21) which is independent of Ceq and thus of the pressure PB' If kt = 00, -rst will be zero. If ktl ~ övk eq. (20) may he' written in the form and for Co ~ Ceq I Ö Ceq- ie, -rst = In, vk kt Ceq- Co (22) -rst =-. vk (23) Hence if the original concentration Co of the rod is zero, or small in proportion to Ceq, -rst will never exceed the value llvk. In principle, however, as may be seen from (20), -rst depends on ölkt and this value is a good measure of the magnitude of -rat, provided ktl is not too small compared with Mv. 3.5 Other stationary concentrations For practical reasons the vapour pressure is limited between certain values. Therefore, because v and 1 are also limited, and because the values of kt and Co are generally small, the Cst that can be reached using optimum conditions (the greatest values of PB and l, and the smallest value of v) sometimes does not differ much from Co' If the value of Ceq at this optimum pressure is high enough, it is still possible to reach a stationary concentration which differs more from Co than that treated in section 3.4. In fact there is a range of such stationary concentrations. In the first place, we can try to give Co a higher value. This might be done in melting together A and B under a high pressure of an inert gas. It is very difficult to make a homogeneous rod under these conditions, however, because "normal freezing" will take place during the solidification. If k; is not too small but kt/öv is 'small compared with kil, then -rst, which we will call -r~~ in the following, has a not too large value. Let us take the rod of material with a homogeneous Co and melt the first zone under the maximal value of PB and wait a time -r~~). After that time the

9 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 327 molten zone with length I and velocity v is passed through the rod; a rod is obtained with a concentration C~~ which is homogeneous between x = 0 and x= L'_l_ When the last zone length I is molten, the movement of the molten zone is stopped for a time.~~. After that time the zone is moved backward with the same speed. Provided.~~ has the right value, a new concentratien C~~is now obtained which is homogeneous between x. I and x = L. It is easily seen that "Co" in this case is C~~ and hence the value of C~~is (24) If this treatment is repeated n times, waiting a 'time.~~) at every turn when the first or the last zone length I is molten, the final concentration reached is (n) ( r5vk )n ( ) Cst = r5vk + kti Co- Ceq + Ceq. (25) Or if we call C~~)- Ceq = Lln and Co - Ceq = Llo, r5vk n Ll n = (r5vk + kti) Llo (26) Lln represents the difference between the nth stationary concentration and the equilibrium concentration. If we want to make this difference smaller than a certain small value y, the value of n required may be calculated: log Llo -log y n~------~~--~~---- (27) log (r5vk + krl) - log r5vk The value of.~~) mentioned above is (n).st = In ( 1+ - kti). kt r5vk (28) This holds for every n except for n = 1. In that case we obtain formula (20), but for Co = 0 we obtain (21) which has the same value as (28). So we may say that under given conditions (15, 1, V, k; and k) 7:~~) is a constant, independent of n and PB' except if Co =!= 0, because in that case 7:~~ has a different value and depends on PB' We give two numerical examples (the concentrations are given in atoms/cmê): (a) k = 0,9, k; = 10-4 cm/sec, I = 2 cm, v = cm/sec, 15= 1 cm, Co = 0, Ceq = 1015/cm 3 ; C~~)= - (0'90)n 1015/cm /cm3,

10 328 J. van den BOOMGAARD C~J = /cm 3, C~~ = /cm 3, C~~ = /cm 3, C~~ = /cm 3, 7:~~) = 1060 sec = 17 min 40 sec. C~~o)= /cm 3 (b) k = 10-1, k; = cm/sec, l = 2 cm, 0 = 1 cm, v = cm/sec, Co = I0, Ceq = /cm 3 ; C~~)= - (0 29t /cm /cm 3, C~~ = /cm 3,. C~~ = /cm 3, 7:~~) = 1 h 23 min 20 sec. C(5) st = /cm 3 If v and l are also changed, it is possible to reach stationary concentrations lying between the mentioned ones. It may be asked, what will happen if the zone is not stopped for the prescribed times 7:~~) but is moved backward immediately after the last zone length l of the rod is totally molten. The answer to this question can be given by calculating the concentration as a function of place after every to or fro course for 7:~~ = O. The differential equation of the nth pass is ~C1n) + (!!.. + kr) C(n) = k r C +!!_ CI';-I) dx l ov x ov eq l 1-x (29) in which L' = L-l and C}_';_-;;I)means the concentration after the (n-l)th pass at place L'-x. The coordinate x is always chosen in the direction of v. The boundary condition of (29) is Equations (n) _ C1n-1) C o - L' (30) (29) and (30) are valid for every n ~ 2. For n = 1 holds dc -) (k kr) (1) kr k ~ + T + ov C x = ov C eq + T Co with the boundary condition (according to eq. (14)): (31) kr (1) W --T CO = Ceq - (Ceq-kCo)e /i st, (32) which gives ct 1 )= kco for 7:~~ = O.

11 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 329 With eqs (29) - (32), cf:) can easily be calculated from C~); C};) from C~2) and so on. So we are able in principle to calculate the solution of the nth equation. A qualitative picture of C~n) as a function of X with n as a parameter is shown in fig. 5. ex f 1 1 I 1 I.1_ 1 1 I 1 L Fig. 5. Qualitative course of ex as a function of x and n. When n is getting large, however, the stepwise calculation of C~n) becomes rather laborious, but an indication about its behaviour may then be obtained in the following way. Dr Bremmer and Dr Braun 5) of this laboratory have shown that the quantity Lln(x) = IC~n)_ Ceql has the following property: -'l~ 1 Iim r Lln(x) = -, n-+'" Ä. in which Ä. is the smallest positive root of the equation Here the dimensionless quantities and (W + a- f3ä.2) e W - Ä.(W + a- (3) = O. krl' kl' a=--+-,!5v l a and f3 are given by f3=-, kl' l (33) From the definition of Ä. it follows that l~) ~alf3 and that À increases with increasing krl'löv and with decreasing kl'll. Formula (33) gives an indication that the number of courses n necessary

12 330 J. van den BOOMGAARD to reach the equilibrium concentration will be smaller for larger Ä. Taking for instance L' = 20 cm, 1= 2 cm, ~ = 1 cm, v = cm/sec, k = 10-3 and k; = 10-6 cm/sec, we find Ä = 1 0n. With the same values for L', l, ~ and v, but k = 10-1 and kr = 10-4 cm/sec, A = Thus the number of to and fro courses to reach Ceq will he much smaller in the latter case. 3.6 The distribution of the concentration in the last zone length 1 Up till now attention has only been paid to those cases in which the molten zone did not leave the rod. In practice, the experiment is stopped after one or more passes of the molten zone which thus has to leave the rod. This will happen for instance at the moment when x = L'. A new variable y = x-l' is chosen and the problem is calculated in terms of y (see fig. 6). In the same way as was done in section 3.2, the three contributions in the change of ry are calculated. The variation of the concentration due to crystallization is (34) Y-V Solid Liquid Cy 1 o-y l-y Fig. 6. Solidification by "normal freezing" As there is no solid at the other side which will dissolve, no material enters the liquid from this side. Therefore The change III concentration resulting from the reaction with the gas phase is k r d3ry = ~v (req- ry) dy. (35) With the aid of (34) and (35) the net change in T y is found to satisfy dry (k-l kr) kt dy + l-y + ~v ry= ~v r eq, (36) or as Cy = T.r y, ac, ('k - 1 kj') kr Cy=-Ceq dy l-y ~v ~v (37)

13 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 331 The solution of (37) is (38) The remammg integral is the difference of two incomplete gamma functions, which can be calculated if the constants (J, I, k; and v are known. The value of Ci, for a case in which the zone leaves the rod after the nth course, will be Ci = ci~).for k; = 0 and n = 1 we obtain from (38) ( l_y)k-l Cy = cf.) -l-. (39) This is the formula for traditional "normal freezing". If the rod of A has a length l and a concentration Co of B, and if it is crystallized from one end, the concentration Ci is kr --'t" Ci = çeq- (Ceq-kCo) e d, where 7: means the time between the moment the rod is totally molten and the beginning of the freezing. If in this case k; = 0, then Ci = kco and (3.8) changes into. I k-l (41) Cy = ie; ( I Y). (40) By putting l = 1 we arrive at c; = ie; (l-y)k-i, (42) which is the formula given by Pfann 1). 3.7 The ultimate distribution after a large number of zones have passed in one direction To calculate the ultimate distribution of B in A after passing a very large number of molten zones in one direction through the rod under a constant pressure PB of B, an extra approximation has to be made. A formula will be derived which cannot apply in the last zone l, where "normal freezing" prevails as described in section 3.6. The effect of the distribution in the last zone is reflected back into the preceding zones; according to Pfann, however, this effect is small, especially at the first end of the rod. When the ultimate distribution is reached, the differential equation of the last course will have the same solution as the differential equation of the preceding course. This differential equation is nearly the same as that derived in section 3.2. The only difference is that the change of concentration in the zone by melting a volume dx of the charge is not Co dx but dx

14 332 J. van den BOOMGAARD multiplied by the concentration at the place x + l; This concentration resulted from the passing of the preceding zone. As the passing of the last zone does not change the concentration, we have to write C x + I instead of Coin (11) and we obtain d c, (k kr) k i; dx + T + ov C x - T Cx+l = ov C eq. (43) A possible solution *) of (43) is A B,x C C A + X - eq = 1e BIX 2e, (44) whcrc BI and B 2 are the two real roots of the equation k BI k kr -B+-e =-+-(Bl>O, B2~0). 1 1 ov The values of Al and A2 depend on the boundary conditions of the rod. For a semi-infinite rod the value of Cx f~r x = oo is Ceq: This is only possible if Al = o. The value of A2 may be calculated with the aid of the boundary condition at x = O. If all the zones passed through the rod start at a time i after the melting of the first zone length l, A 2 must satisfy the following relation: S k Jl l _!::.T C(O)=A2+Ceq=Ceq-?Ceq-T CxdxS e rj where C(O) is the value of Cx for the ultimate distribution at place x = o. Hence: o (45) (46) If the rod has a finite length Land kr =l= 0, Al and A2 can be found with the aid of the boundary condition at x = 0 and the condition L J Cx dx = L Co+ R o (47) where R is the amount of B dissolved in A from the reaction with the gas phase, but this quantity cannot be calculated. If k; = 0 it can easily he proved that A 2 and B 2 are both equal to zero and the formula of Pfann is obtained. *) The problem of finding the exact general solution of eq. (43) is very difficult and has not yet been achieved. The solution discussed here is analogous to that proposed by Pfann 1) for the case k; = o.

15 ZONE MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE The dependence of Ceq on the vapour pressure of B In the previous sections the concentration Ceq of B in the melt in equilibrium with a vapour of a certain pressure PB was introduced, without considering the dependence of Ceq on PB' The form of this dependence depends on the model we use for the liquid. If B dissolves as atoms (as assumed in section 3.1), the equilibrium gas-liquid can be described by the reaction Bg ~ BI (a) and application of the law of mass-action shows that. [BI] = KlP B (b) If the vapour contains molecules like B 2 rather then atoms, the dependences change accordingly and (a) becomes and thus l B 2 :t- BI [BI] = Klpt,. (c) (d) In many semiconductors B forms a centre that easily splits off an electron (if B is a donor) or a hole (if B is an acceptor). For a donor centre (e) and thus [el [Bn = K. (f) [BI] 2 At temperatures near the melting point the ionization in many cases is practically complete. Accordingly (a) can he rewritten and application Bg~Bt + e of the law of mass-action now leads to [Bn [el = K 3 P B If the free electrons present in the liquid are mainly due to donor ionization: and hence Many molten semiconductors behave either as metals or as a semiconductor with a small energy gap. In this case there will be a considerable concentration of intrinsic electrons and holes at the temperature of the melting point. Accordingly (g) no longer holds, but in (f) [el is a constant and therefore the concentration of B is again proportional to PB' (g) (h) (i) (k)

16 334 J. van den BOOMGAARD In cases in which the ionization is partialor in which the concentration of intrinsic electrons is of the same order as that of the centres, a more general treatment along the lines indicated elsewhere 6) is neces!?ary *). 4. Comparison of theory with experiment Through a rod of n-type Ge (70 Ocm) a molten zone was passed with a length of 2 cm and a velocity of cm/sec. After 3 cm the zone was stopped and Se pressure of 3 mm Hg was applied. The zone was then moved over another 3 cm with the same speed. After that, the zone was stopped again and the Se pressure was raised to 40 mm Hg, and after that the zone was moved over the last 4 cm. By putting a seed crystal before the rod, a monocrystalline rod of Ge was obtained, containing three different regions. After etching the surface, the whole rod was tempered at 600 C and the resistivity was measured by the 4-points method at 25 C. The first region was nearly intrinsic. The second zone still had a rather high resistivity and was n-type. In the third region, which also showed n-type conductivity, the resistivity had a much smaller value, and these values could be used to check the theory, because disturbing influences are relatively small here. The course of the conductivity ax (which is the reciprocal value of the resistivity) is given by the experimental points of fig. 7. If the distance between the donor levels of Se in Ge and the conduction band depend only slightly on the concentration it may be said that and so from (12) in which axcsl ex (k kr) ( --+-x (k kr)) --+-x ax = ai e I 6v + ast 1- e I 6v, (48) ax = the resistivity at any x, ai -" " at x = 0, ast " " extrapolated for x = 00. Formula (48) can be written in another way:. a' lolog(ast-ax) = lolog(ast-ai) - - x 2 3 in which a' = kil + kt/~v. (49) *) The same holds for the distribution equilibrium. If nearly all donors in the liquid as well as in the solid are ionized and the concentration of electrons in both phases is independent of the impurity content, the distribution coefficientmay be defined as k [B~lid] B if'h.... ial h. f h. trins. = ut t e tomzatton IS parti,or t e concentratten 0 t e III IC [B liquid ] electrons smaller or of the same order as that of the centres, the more general treatment is to be used.

17 ZONE-MELTING PROCESSES UNDER INFLUENCE OF THE ATMOSPHERE 335 Iiil Experimental points t-- Experimental points used in fig.8 - (J'x,.74xlo-le-O.5x+3xIO-1 (I_eO.5x j.n;-7, m-7 t-- ct;-1.7'xio- 1.a-lcm-I o;t"jxio- 1.S: b" I 2.0 / ~Oo~~~~~~~~~-L--~~~ ~ ~ 2 ~ 3 ~ 4 ~ --- x(in cm) Fig. 7. The conductivity of a monocrystalline rod of Ge with Se as a function of place x. In fig. 8 l log (O'st-O'x) is plotted against x. As the value of O'st was not known, it was estimated from fig. 7 to he n- 1 cm-1;this value was used in fig. 8 to calculate the experimental points. The straight line of fig. 8 fits the experimental points rather well. From this line the following values of the constants may be obtained: So (48) may be written as a' = 0,5, which is the curve drawn in fig. 7. O'i = 1, n-1cm-1 O'x = e-o. 5 ", (1- eo' 5 "') The last zone length of the rod was solidified very rapidly. (50) During solidification Se evaporated rapidly from the melt. From this ohservation it may be concluded that k ~ 1 and so kil ~ 1- as l was 2 cm. As a' == (k/l+kt/ c5v) = 0,5, k; must be of the order of 0 5 c5v and from this, with v = 2.10-~ cm/sec and c5 = 1 cm, kt was calculated to be of the order of 10-3 cm/sec..

18 336 J. van den BOOMGAARD Experimental points IOlog(o;t- o:)='olog( Cf,- o;~-s- f-,-- O:t-3xIO I-IOiiit- tg(3~ _...1:L (CÎst-c1i) ::-: b" I -1.0 ~ ' ~i D,-Icm-I ; '.95 oe:" 2:;~' -0.5cm- 1 1'0 I ~ '" r-, '\ r-, "- I\. -, l\.. -, I f-- I-- -f.4. -: <, 9 r-, -, , x(in cm) _ Fig. 8. lolog(ast - a",) plotted against x. The resistivity values at the side of the rod which faced the atmosphere during the levelling process and at the side which faced the container during that process were found to be identical. So the assumption of section 3.1, sub b seems to hold rather well. Acknowledgment The author is greatly indebted to Dr F. A. Kröger for many stimulating discussions and to Dr H. Bremnier and Dr G. Braun for their aid in solving the differential equations.. Eindhoven, August 1955 REFERENCES 1) W. G. Pfann, J. Metals, N.Y. 4, , ) W. G. Pfann, J. Metals, N.Y. 4, , ) H. Reiss, J. Metals, N.Y. 6, , 195'4. 4) J. van den Boomgaard, F. A. Kröger and H. J. Vink, Brit. J. Electronics 1, , ) The result of these calculations will be published elsewhere. 6) F. A. Kröger, H. J. Vink and J. van den Boomgaard, Z. phys. Chem. 203, 1-72, 1954.