pm ‘ fl 3. O ’3 0.5 1 “git“ has" n‘ _ . . C 1% I: . KI...“ . .tm _. v I O 3 as b a u.» ‘ x . I‘ r....\~ ‘3 F~ kmw k s ".7 B 'I? ”w I.” v “figm- o‘t. . V 1.3:. rt, ’3.“ amu t. a”. y . .. w F. max. 3.. .1. R f . 3.3 y . “1.; u. f. h“. . {.0 “a. s3 3 a. fl v a s - a.» \. :- ¢ 52-2.: 3. K . _ A 1 W. flaw Mu“... H_ :1 E; :2: " «1mmmnnmmn'mm mm! “ 3 1293 01743 0343 . k: I" LIBRARY i MM Ezigjjan St;th , a L Unrxrsxtu] v...— CONDUCTIVITY OF THIN METALLIC WIRES By Helmut Gerhard Satz ’AN ABSTRACT Submitted to the College of Science and Arts Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1959 Approved M fr/l [Mr ABSTRACT +) Measurements by Olsen on thin indium wires at low temperatures have demonstrated that not only the residual but also the temperature dependent part of the resistivity increases as the wire diameter decreases. It was suggested by Olsen that small angle phonon scattering, which may take electrons to the surface, where they suffer diffuse scat- tering, might give rise to the observed effect. Since an exact solution of the transport equation in this case is beset with nearly insurmountable difficulties, we have re- sorted to an extremely crude analysis similar to that em- ployed by Nordheim++). Two mechanisms are considered: 1) Electron-phonon scattering, which takes electrons to the surface in a time shorter than 1;:h , where 1:3h is the electron-phonon relaxation time in the bulk. 2) Electron- electron scattering, which, although of no consequence in the bulk, may also contribute to the resistivity of a thin wire by bringing carriers to the surface. Both processes lead to a size and temperature dependent contribution to the total resistivity of thin wires, although in the tempe- rature region considered here the size dependence of the electron-electron effect is very weak, and thus this pro- cess could account for at most a small part of the observed resistivity. +): Olsen, J. L., Helv. khys. Acta, 21, 715 (1958) ++): Nordheim, L., Act. Sci. et Ind., No.131 (PariszHermann) CONDUCTIVITY OF THIN METALLIC WIRES By Helmut Gerhard Satz A THESIS Submitted to the College of Science and Arts Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1959 ACKNOWLEDGEMENT It is my pleasure to thank Professor F. J. Blatt for suggesting this problem and for his continued assist- ance and encouragement throughout the course of the work. The financial support of the U. S. Air Force, Of— fice of Scientific Research, is gratefully acknowledged. I. II. III. IV. V. VI. TABLE OF CONTENTS Introduction.......................................1 Outline of the Strict Transport Theory Analysis....9 Kinetic Approximation.............................15 Discussion and Comparison with IXperimental Results...........................................23 A. Temperature Dependence.........................24 B. Size Dependence................................27 C. Electron-Electron Effects......................28 D. Comparison of Various Mechanisms...............29 Conclusion........................................51 Appendix..........................................55 I. Derivation of the "Distance of Flight" for Electron-Ihonon Scattering...................55 II. Outline of the Multiple Scattering Proba- bility Derivation............................34 III. Consistency Check: Bulk Thermal Resistivity via Multiple Scattering Method...............56 BibliographyOOOOO00-000....COO...00....00.000.000.00000057 I. INTRODUCTION If one attempts to explain the electrical properties of metals on the basis of an essentially free electron gas obeying Fermi-Dirac statistics (Sommerfeld theory), then the electrical conductivity is obtained in terms of the electronic "mean free path" 2 as CT = [mi-H n1U' where 63 and n. are the electronic charge and the number of electrons per unit volume, respectively; Elia the velo- city of an electron at the surface of the Fermi distribu- tion, and "1 the mass (or effective mass) of the electron. Since 3' remains quite constant over a very wide range of temperature from O°K upward (the electrons forming a strong- ly degenerate Fermi gas), the above relation gives the con- ductivity, aside from constants, in terms of only the mean free path.1) At normal temperatures this mean free path is very much shorter than any dimension of a specimen, and hence the conductivity in that temperature region is independent of the size and shape of the conductor. As the temperature is decreased, however, the mean free path in a pure metal increases, due to a diminished number of thermal vibrations (phonons) present at lower temperatures, and hence for thin wires and films at very low temperatures the dimensions of the conductor may be comparable to or smaller than the mean free path, thus suggesting that under these circumstances there should be a dependence of the conductivity on the geo- metrical shape of the conductor. Experimentally it is in— deed foundg) that at very low temperatures the conductivi- ty of thin wires and films decreases with decreasing dia- meter or thickness. This result can be interpreted as an additional resistivity mechanism arising from an increased importance of electron collisions with the surface of the conductor, assuming these collisions to be at least partial- ly diffuse. In calculating the low temperature conducti- vity of thin wires or films the problem then arises as to how these additional surface effects are to be treated. The major theoretical investigations of this question were carried out by Fuchsa)and Dingle“); both authors give critical evaluations of the work done previous to theirs. A brief review of Fuchs' method for thin films will now be given here; Dingle's work consists essentially in extending this procedure to a cylindrical geometry. In Fuchs' treatment the following main assumptions are made: 1) Electron scattering at the surface is completely dif- fuse; i.e., an electron is scattered from the surface into any solid angle with equal probability, independent of its initial direction of motion. (Fuchs also treats the case of partially diffuse, partially specular scattering; expe- rimental resultsS) seem to indicate, however, that such scattering is largely diffuse, and hence only this case will be considered here). 2) In the bulk material the probability per unit time that an electron is scattered through an angle 69 is inde- pendent of<3 ; this probability may, however, be a function of the electron energy. These assumptions state in effect that any collision suffered by an electron will completely randomize its momen- tum, independent of the initial direction of motion. A transport equation for the disturbed distribution function {(3,2) can then be written, in which the integral operator for scattering in the bulk is replaced by a “relax- ation" term involving a relaxation time't ; this is justi- fied by assumption 2) above. (For the geometry, see Figure 1-1). The non-equilibrium distribution function {(irfiz) is now expressed in terms of the equilibrium (zero electric field) distribution function {o<fi?w and a small perturbation term {Anna : {(3,2) = {0(6) +£0.73) With this, the transport equation becomes, to first order, a partial differential equation for ¥A(J{z) : 914 + ":1 =- E QJEO (I‘ll) V I} 2 rum Z”"”"’““"“*';* Squus ‘ i E 2- .4»................. --..--._...-. --- - - ...-—.....-.-..--.-......) x Figgre I-1)A Equation (1—1) is then solved, subject to the following boundary conditions: At 2 = O,{¥Zz)is independent of the direction of E? for all 02>0, and at z = a,f4(6‘:2) is independent of the direction of L? for all 0£< 0; these conditions are equivalent to the assumption of perfectly diffuse scattering at the surfaces. The solution of (1-1) is used to find the mean current den- sity and hence the conductivity: J= v E = ffimaf‘az’““xd‘addz] “'2 O The results thus obtained can be written in terms of ele- mentary functions only for the limiting cases of thick and very thin films, i.e., for £<>q , where I is the mean free path and (L the film thickness. The results are given below, together with the correSponding expressions for wires, as found by Dingle4): FILMS: E1335" + 1 0000000000000000 K>>‘ (1-2) 0‘ 8K 3 a f:— l l J ooooooooooo I- 0" 3K [ 403‘?) K «I ( 3) where K: a/f ; U; and a, are bulk conductivity and film thickness, respectively. WIRES: _ ' .1. i ooooooooooooooo K>>\ (1-14) cm WIRES, continued: ___. = l/K oeooooooooooeeooooooooooooo K<.. Cb .+ Eb (Matthiessen s Rule) Considering now the results of the theory for wires (since Olsen investigated indium wires of various diameters over a low temperature region), one finds for the resistivity (of. equations I-4 and I-5): For K»! , (3w: (Db 4' 3/4[(3be/0‘:| RS5 = f». d'+ (33% + 3/4 [eta/0.] g; (abttiid + (3;ij + Coughu+x( l/d) and fori<?<§23HE"— {majwamr' (11'3” Here ?(5’:u)=1’(5’.3') is the transition probability per unit , i.e., -I do time from statel} to v and vice versa. Introducing now the perturbation expansion for'f , equation (II-2) becomes -11- Sf: 3{, = - M672) “I -— it?) '3) }"\’(6’v (7').)olt7" (II—5) If one now again assumes that in the bulk material there exist two statistically independent scattering mechanisms, one due to stationary scattering centers, the other due to phonons, then P:?I+FT‘ where I): is the impurity and F;- the phonon scattering tran- sition probability. With this substitution equation (II-5) separates into 3A : S)« { [f{\"f‘/{} “I (10' ”jib 4/{1§?0{J](H"4) where gzfipk), etc. It is noted that this possible separation, i.e., the statistical independence of the two types of resistance me- chanisms, does 333 imply matthiessen's rule; the require- ments for the validity of the latter go considerably further; namely, it has to be requiredg) above and beyond (II-4) that 1) unambiguous relaxation times can be defined for both types of processes; 2) the ratio of these relaxation times is independent -9 of LI . Requirements 1) and 2) are contained in the assumption made in the Fuchs-Dingle treatment, that a relaxation time I: can be defined for the total of all resistivity mechanisms, independent of the scattering angle and of the electron's initial direction of motion. In the general equation (II-4) neither of these two conditions is necessarily fulfilled. It will, therefore, be necessary to investigate more explicit expressions for IE and F} . Assuming static imperfections, P;- can be written as Egg») 3 'Pl(e) = [tr/Q] I (9) (11-5) .5 ~)' Here 6 is the angle betweenu' and U’ , the initial and fi- 10) nal velocities of the electron; .0. is the atomic volume (assuming one conduction electron per atom), and 1(9) is the scattering cross-section of the particular type of imper- fection. Similarly PT can be written“) 2 Rr (3"?) s 7’? (9) s a, C [w;]'§(ec+§_€f-twq’)(H—6) ’ M3 4- azC [LMNAL U] JOSE»? " 6‘: ‘* AW?) Lou?) Here 0.4 and QL are constants; C is the electron-phonon coupling "constant"; g’is the phonon wave vector with the corresponding frequency'buCih); Adi is the number of phonons with wave vectors a? ; finally, 6: is the election energy, corresponding to an electron with a wave vector h: (propor- tional to J? ). The first term of (II-6) corresponds to absorption, the second to emission of a phonon. From these expressions it is clear that in general both 1%: and 12. are dependent on a? , and moreover, that this dependence in general is different for ‘PI and 1?}- : in P1 -15- it is temperature independent but varies with the type of impurity, whi.le in 7%. it is strongly temperature dependent, in particular through the dependence of N? on temperature (Planck's distribution). Moreover, a relaxation time for electron-phonon scattering can generally not be defined for temperatures much below the Debye temperatureqa). In a correct analysis it is therefore not permissible to define a relaxation time including all bulk processes at the outset, since requirements 1) and 2) (p.11) are probab- ly not satisfied. In order to solve the problem exactly one would thus have to solve, without any further major approximations, the transport equation as it stands: -;2%2{,(&°.2)+ 9;? ifim)=_194(5’,2) X (II-7) X - {Ll-€13) .9 a” .7! 4073;) Q- I I i j [I MRHlE’w'U )alv +j[| - {I.zatahw'flfl} where T}. and '9} are given by (II-5) and (II-6); equation (II-7) is then subject to the appropriate boundary conditions of completely diffuse scattering at the surfaces. The strict analysis will not be carried beyond this point, for the following reasons: Equation (II-7) is an integro-differential equation in which the kernel of the integral operator, among other things is dependent on the boundary conditions. Its solution, if at all possible, would be mathematically very difficult, if no further simplifying assumptions can be made. A solution -14- has been obtained by Sondheimerqa), but under the very spe- cial assumption that 727(9) = a Coflf9. What will therefore be attempted instead is the follo- wing. Abandoning a rigorous mathematical procedure based on transport theory, we shall try to construct a very crude picture of the processes involved, based on simple physical models, and try to see whether it is possible to obtain re- sults giving at least rough overall agreement with experi- mental results. That such an approach may not be totally worthless is perhaps supported by the fact that a very crude "kinetic" treatment of Nordheimqu) gives results which for 15) thin wires agree to within 5% over the entire range of!( (i.e., of various diameters at a fixed temperature) with the "exact" results of Dingle's treatment. One may therefore hOpe that a crude treatment based on simple physical models will at least give some insight into the role played by se- veral scattering mechansims not considered by Fuchs or Dingle. -15- III. KINETIC APPROXIMATION. At the outset we shall classify the various resistivi- ty mechanisms present in thin wires and films as follows: A) Bulk mechanisms (momentum randomized without surface interaction) 1) Scattering by lattice imperfections --- tempera- ture independent. 2) Scattering by phonons --- temperature dependent. B) Surface mechanisms (momentum randomized by complete- ly diffuse scattering at the surfaces) 5) Simple surface scattering --— temperature indepen- dent. 4) Small angle phonon scatterings which bring the electron to the surface --- temperature dependent. 5) Electron-electron collisions which bring the elec- tron to the surface --- temperature dependent. The bulk processes are assumed to be known and are given in terms of mean free paths: RI : due to imperfections (residual bulk resistivity) {T : due to phonons ( ideal bulk resistivity) The simple surface scattering (residual surface ef- fects) can be approximated roughly by a simple argument as followsqn): Considering an electron starting at the center of the wire at an angle 69 with the wire axis (see Figure III-1), we obtain a mean free path!SS by simply averaging over -16- all directions+ "' 3 "' Or [>33 ‘ O( 1‘ ~‘ 0< 16w5+M4+ <——7 f X ‘- - '—' ***** Wirt QAI'J Figure III-1 It may be noted here that the result of our crude picture gives an answer for residual effects I I \ l '1; 3 w + J- : —-- + 0" (“I—I) - ( DU : constant; d : diameter) - which agrees essentially with Dingle's rigorous result Ldk : (Drain-l. I (greed. “" (51:61:13.2: /a‘] ( ('5 : constant). We now wish to find a mean free path Etsfor the resisti- vity arising from electrons being brought to the surface by phonon interactions. It should be remarked here that the separation of the mechanisms listed under 2) and 4) above (p. 15) is here assumed possible, even though these processes are really not statistically independent. M +): For the objections to this procedure in a rigorous treatment, see Fuchsa). We have essentially averaged over all paths of one electron rather than over all electrons at a given moment. We now calculate the desired 0T3 as follows. at low temperatures each individual electron-phonon interaction produces scattering through an extremely small angle €>16): 9 £- 9“, a #93 where (hp is the Debye temperature. For larger angles the transition probability rapidly becomes vanishingly small. Thus many small angle scattering events will be necessary to bring an electron to the surface if it is initially mo- ving along the wire axis. The average distance travelled before each collision (inversely proportional to the transi- tion probability) is given by17) (see Appendix, II.) a 2 (KT)“___ __|_ [Sne‘flaflsd' (3.? where c: is the velocity of sound, n. is the number of elec— trons per unit volume, and PI is the thermal bulk resisti- vity. We shall now assume that this distance is a constant for all scattering angles from zero to (3wmx; the problem then requires finding the mean free path for an electron undergoing multiple scattering such that 1) each individual event causes a scattering through an angle Q4 9.”; 2) the average distance of flight (and hence also the inverse of the transition probability) between two events is taken to be a constant (<1 ) over all angles from zero to<§wm , and infinite for all larger angles; 5) the electron, after having undergone a large number of such collisions, is brought to the surface, where its momentum is randomized. Starting with this information, Ens can be obtained (in a not totally unambiguous fashion) by a method developed in cosmic ray theoryqe). Consider the wire axis to be the x- axis of a Cartesian coordinate system (see Figure III-2); the radius of the wire is r . Then the probability that 4t éélcctrou Pa“! r ’\ 13* /\ ’ , ... ’ ‘ (112/ (rp- --— ‘ " : 7x~— .- \ XI v /" s \ \f ./ Figure III-2 an electron, initially moving along the wire axis, reaches the surface of the wire through multiple scattering after having traversed a distance >< along the wire axis (of. Fi- gure III-2) is given by (see Appendix, II.) - _ EU]- "'-'3/ ’3 L1-..5 HOW) - [2»? X LGXPE ZLJr x} where W: [ZR/d. 9L1 jl/z " auax CL being the "distance of flight" and Bumpthe maximum scattering angle as defined above. We now define as the mean free path (43 of this pro- cess that distance SEQ along the wire axis, for which the probability that the electron has reached the surface is a maximum. That is to say, we determine the distance {galong the wire axis by maximizing HG?) (the radius r is fixed). This procedure is ambiguous to the extent that the mean free path need not necessarily correspond to a maximum of HC?) - one could equally well ask that the probability of the elec- tron reaching the surface have any particular fixed value, such as, for instance, 1/e; furthermore, the mean free path is actually the path taken by the electron in going from the wire axis to a point on the surface --- not the correspond- ing distance along the wire axis. The first point, however, presumably does not intoduce any fundamentally significant difficulties+), whereas the second can be justified by as- suming a sufficiently thin wire, so that the two distances are approximately equal. Finally we shall assume that the mean free path thus determined is approximately valid for all electrons (i.e., also those not initially moving along the wire axis). This then allows the most simple mathema- tical treatment. Maximizing HQ?) with reSpect to Y we find l 1 2/ § _ [MRI/3"“ Zarl ]/3= [.IV‘LJQ [.3 ° F 92,, 41: VIC" ((915)?! where p: was the thermal ("ideal") bulk resistivity and f the wire radius. We thus have as the approximate mean free path for thermal surface scattering +): Applying this procedure to thermal bulk scattering does result in essentially the correct ideal resistivity. See appendix,III., where this consistency is shown. -20- vmr V3 V343 €75 '- [#111461] ((3313 Qualitatively we note that '/{13 , and therefore the resisti- vity arising from this mechanism, is temperature dependent (through ((‘J:)V3 ), and that its contribution is greater for thinner wires, as one would expect and as found by Olsen. Lastly we wish to consider electron-electron effects. The mean free path (in the bulk) for electron-electron col— lisions has been found by Abrahamsqg) to be approximately I__..‘.._£-;;’~ ee‘mqgk'r where EF is the Fermi energy and O; is the electron-electron scattering cross-sectione U}. is temperature independent since the Fermi energy remains quite constant over a wide temperature range. We note again that due to momentum con— servation there will be no bulk resistive effects arising from electron-electron scattering. We find an approximation to the mean free path for electron-electron-to-surface processes by a simple argument similar to the one used to find the simple surface scattering mean free path. Consider an electron moving along the wire axis; after a distance fee it will experience an electron- electron collision in which the total electronic momentum is conserved; the mean distance 5 , which the electron now still has to traverse in order to reach the surface (where its momentum is randomized), was found before (for st ) to be qn/W', r being the wire radius (see Figure III-5). -2, '1 - hence here the total mean free path for the entire process in a thin wire is roughly £€ no; KT TI (KT) “0?: H Again we note qualitatively that UQse and thus the resisti- vity due to this process increases as the wire radius is decreased; we also see that in the bulk ( I” --> OO ) the ef- fect vanishes, as expected, and finally, that the mechanism becomes more important as the temperature is increased. Combining all our results, we find for the total mean free path in thin wires —L:: "" + -' '1' -- + '+ | {w (I (T egg (TS o i I 2‘ J. + J. _,. If, [QTYNCJI5E7/3+ :(K) e: {T ”T er r bF/an I L—g-WKTI Thus, for the thin wire resistivity, using the Sommerfeld l E’ (111-1) H +- theory expression Wit -22... we obtain the result w = PI He: :+]-‘; (Him—5]” A...) (In—2) U‘ _(K7 + [fi,][ ENG} + “HKTV/n :1 L It where (>1 and (>,. are the bulk residual and ideal resisti- vities, respectively. This can be rewritten in a simpler form | 2.. a b . 5 . I (I 3< 3)”+ ”m (III-3) (’W (’I (’7 AL: \LMIA) €37: ES/uo; , “KIWI,” where A =[-ne‘] is, for a given metal, a temperature and .v size independent constant. IV. DISCUSSION AND COMPARISON WITH EXIERIMENTAL RESULTS We had obtained the following expression for the re- sistivity of thin wires _ b t 1_ AVV3(T)5 (KTAL ] _ (aw-51+(7T+A[4r +(~) I)... + )L (III 5) A rag Eé/m; + “gm Briefly looking at the qualitative behaviour of (III-5) one sees that 1) for the bulk material, i.e., for V'——>, each term within the square brackets vanishes and we obtain the cor- rect bulk resistivity; o b 4* . 2) at T s O K all terms except ft. and pgll vanish and we find the experimentally satisfactory Nordheim result .~ 0 e QA)_L Pw(‘“o ‘0 = (’5. + A?) r 5)the last two terms within the square brackets are tem- perature as well as size dependent, in contradiction to a strict matthiessen's rule, but in qualitative agreement with Olsen's results; 4) the size dependent part of the resistivity increases in importance the thinner the wires are made, and the total resistivity increases with temperature faster for thinner wires. Hence we see that at least the qualitative behaviour of our result gives agreement with Olsen's eXperimental findings; we shall in the following attempt a somewhat more quantitative comparison. ’2 -::4- A. Temperature Dependence Te shall consider here the temperature dependence of Pw,without, for the moment, taking into account any elec- tron-electron effects. Then we have I1 2I V _ O s @t)3A3 b 3 (Dw() 2' PW - ()WCO K) *- (be? 2 (3T 4» I * (ii/5C?! J (IV-1) The value of (\ is determined from Olsen's experiments, by extrapolation of his measurements to T a OOK. We take for ft. the measured resistivity of the heaviest wires minus the extrapolated resistivity of these values for T s OOK. These wires exhibit approximately bulk behaviour. In Figure IV-1+) both the calculated and the experi- mental values of (JWCT) are plotted as a function of tempe- rature. It is appearent that the calculated values are considerably too large and exhibit a slightly incorrect temperature dependence. These discrepancies are possibly due to the following reasons: 1) we have considered only so-called "normal" electron- phonon scattering, which at low temperatures results in on- ly small angle scattering of electrons. There exist; how- ever, even at low temperatures, the possibility that the momentum of an electron is randomized through large angle scattering caused by direct interaction with the lattice +): Figures IV-1 to VI—b are found at the end of Section IV. -25- ("internal Bragg reflection"), i.e., via so—called "Umklapp"- processes. It has been calculated by Bailyn and Brooksgo) that in bulk sodium at 40h Umklapp-processes give rise to about 80 % of the total resistivity, i.eu, eu/puazs . It is almost certain that in the trivalent metal indium Umklapp- processes are at least as important as in sodium, the metal which most nearly approximates free electron gas behaviour. In all probability (JUL/f” >>5‘ for indium. Since, however, in the derivation of the last term of (IV-1): {man‘s A“5 €21} (pg/3 (IV-2) the bulk resistivity was assumed to be due to normal (small angle) processes only, the use of the total experimental bulk resistivity in the evalutation of (IV-2) must lead to" a considerable overestimate of that term. 2) In the derivation of (III-5) the statistical indepen- dence of thermal bulk and thermal surface effects was assumed. This is undoubtedly not valid and may cause an error both in the values and in the temperature dependence of PWCU . We now introduce a multiplicative parameter 0< in ex; pression (IV-1) to account for these effects, i.e., we as- sume that ._ , _ V (W) = (a; + o< {MM/x"! r "33(933 (IV-5) and obtain 0< by matching experimental and theoretical curves at one temperature. We thus find the results shown (for a particular wire) in Figure IV-2. The general comparison between experimental and theoretical temperature dependences is shown in Figure IV-5. These curves were ob- tained as follows: A plot of ENG) vs.:3%3should result in a series of straight lines (of. Figure IV-4), each corre- sponding to a different temperature; the lepes of these lines should, according to our results, be prOportional to (p:)V3; their intercepts should be F: . Comparing the experimental slopes (Figure IV-4) with (€¢)Y3, we obtain Figure IV-B. It is seen that the temperature dependence of the calculated values in general is still somewhat too weak. This is also indicated by Figure IV-5; here a log-log plot of the lepes (~9Ts) vs. the intercepts ( P: ) of Figure IV-4 is shown. According to our results we should obtain a straight line of slope 1/5 (of. equation IV-2); the actual lepe is somewhat steeper, indicating a somewhat stronger temperature dependence of the experimental values than that predicted by our model. A more quantitative statement does not seem justified here, since only a small number of some- what randomly distributed experimental values are available in this case. Of the above reasons for the quantitative failure of the theoretical results, the first could be eliminated, pro- vided the effect of Umklapp-processes on the total bulk resistivity were known. The appropriate changes in the calculations could then be made. To our knowledge, how- ever, there exist at the present time neither calculations nor any experimental data which might allow a reliable +) estimate cd'Pu/Ph, . Such correction, when possible, would however not lead to any significant change in the tempera- ture dependence of PWU), since normal and Umklapp-processes show approximately the same temperature dependence22). It would seem difficult to correct for the second ob- jection, i.e., the statistical correlation of thermal bulk and surface processes (which, as a consequence of the above remarks, would have to account for the temperature depen- dence discrepancies), within the frame-work of our simple model. A suitable parameter (x , as used above, would, if it were made to include effects arising from the statisti- cal correlation of the two types of thermal mechanisms, be- come both temperature and size dependent. The crudeness of our model, however, would hardly warrant or justify such elaborate extensions. B. Size Dependence 4) At T = OOK our results reduce to Nordheim'sq eXpres- sion; Olsen showed that his results extrapolated to T = OOK agree quite well with this eXpression. At higher tempera- tures, according to equation (IV-1), one would expect the temperature dependent resistivity fHJT)to go as pwfi) = (9,“? + {(7) r“: (IV-‘0 +): One could, for example, estimate the ratio PMJFN from the ratio of electrical to thermal conductivity at low temperatures21), if this information were available. A -28- Hence a plot of the experimental values of @JF)vs. rJéshould result, according to our model, in a series of straight lines (each line correSponding to a particular temperature), intersecting the fmufiaxis at the value of the bulk thermal resistivity'ff for that temperature (of. p. 26 also). Such a plot is shown in Figure IV-4; it is seen that the general behaviour of the experimental results agrees with our predictions. However, not enough experimental values are available to reach very definite quantitative conclusions. C. Electron-Electron Effects The temperature variation of the electron-electron-to- surface effects alone can be compared with the temperature behaviour of the various other mechanisms. In Figure IV-6 a plot against temperature of the experimental values for the total thermal resistivity(%fln and the bulk thermal re- sistivity'pt, as well as for the theoretical values for the surface mechanisms is shown. (The latter are matched with (DWFU at 50K for comparison purposes). It would seem rather unlikely, however, that the electron-electron effect, as treated here, is responsible for anything but a fairly small fraction of the total thermal wire resistivity, since the latter does show quite a pronounced size dependence, where- as the electron-electron term, at least in our model, gives rise to a resistivity contribution which is quite insensi- tive to size variations. We are forced to this conclusion by the fact that in our result Pse = A /[ 5;: £293 3-?) (”-5) ‘1 the first term in the denominator (treating hF/Md}. as a parameter) has to be considerably larger thanthe second in order to obtain a quantitatively reasonable contribution form this effect. D. Comparison of Various Mechanisms We see that our model provides a qualitatively reason- able picture of lattice scattering of electrons and the associated surface effects in thin wires; this picture can be improved quantitatively by considering Umklapp-processes. Electron-electron effects, in our approach, seem to give rise to only a lesser contribution, which is quite insen- sitive to size variations. It is appearent that we have made numerous simplifying approximations in the develOpment of our model --- approxi- mations which are not merely difficult to justify, but which may very possibly lead to serious errors. In support of our crude model we can only cite the success of Nordheim‘s equally crude approach. --- Quite aside from the above con- siderations, a number of approximations are implicit in our treatment as well as in all previous work, such as that of Fuchs and Dingle. First we have not considered the effect that localized surface states (Tamm states) might have on the conductivi- ty of thin wires. Practically nothing is known of surface states in metals, although during recent years investiga- tions of this field have been initiatedga). Second, we have assumed throughout, as did all other workers, that a continuum of states in momentum space is available to electrons and phonons. Now it is clear that in a thin wire the number of vibrational modes with wave vectors normal to the wire axis is 5 Na’ where Na is the number of atoms in a cross-sectional area of the wire. For thin wires Na may be sufficiently small so that at low tem- peratures the energy difference between neighbouring vibra- tional modes is of order kT. In that case the usual de- scription (density of states, etc.) for normal electron- phonon scattering must fai124). Size effects attributable to the discreteness of the phonon Spectrum have been obser- ved in the past, most recently by Tanttila and Jennings252 The same boundary conditions which lead to a discrete (not even quasi-continuous) phonon spectrum will similarly also give rise to a discrete electron spectrum, as distinguished from the quasi-continuous spectrum for the conduction elec- trons in the bulk metal. Hence a treatment which assumes a continuous Spectrum cannot be correct.26) 1"JIIV3J« «vi 3'..- vs . In 7" 95 .06mm Haul-9+. V1 I“ L", ¢ .036 mm 57' / “Rodi, 4: __47 \ D (ohm—cm x qu) \ IV: 3' 95 .3” mm “\QOIQ‘R Resushvuiy E» b\ \\\ I» i In 7. eXPt'L. In Lhe‘KP‘f'L In 3, Card] dL + 5 0K Tempered ure Figure IV-1 Experimental and theoretical values for the tempera— ture variation of thin wire resistivity (unadjusted) s6 ixruHL ' WM. m .Sv I; o," p o ,x‘ k . § “meow-t1". J?“ a) x, / Wifl (A) i (Human O ,' >\ .2.» LE 4—- U) R7 Q) (34 .l‘t .l v- |.5 Z 3 4 0 K Tew4perq+ure Figure IV-2 Experimental and theoretical (thermal only) tempera- ture variation of thin wire resistivity for a parti- cular wire (In 4); matched at 30K. Resisfivify (ohm—cm no“) i Experimental. %’ .gi, TheoreHmL .5 + .4 it .3 ‘t Matching T’oin‘l‘ .1 l .I 0 I 2 3 H 5 ‘DK Tempera+ure Figure IV-B Comparison of general temperature dependence, experi- mental and theoretical, of thin wire resistivity (matched at 50K) Resishviiy (ohm-em x IO‘) .7 + TM (HM) T UL1°K) .é + .5 «L- .Ai T(3.5°K) A EM(S.S°K) 3 J .1 .. , T (3.0m) 0 / ,~"/ A/ [/63 ,.// 3M (3.0%) 0/ “ ”K/ o T(2.1°K) ;:oe__——-—o=—r° A ’— 3’Mcz.2°x) to 20 so 40 so cm'Z/3 (oliamehr/z )-z’3 Figure IV-4 EXperimental size dependence of thin wire resistivi- ty, at various temperatures. ( 0 ): Tadanac Brand (T) ; ( £3 ): Johnson,hatthey (,Ihx) 13 uh?! Naif..- nil" fl“. A. ~ shim sum: 22.. cow-cww”) [v Slope of R3. 91 ___' hikeofiku‘u‘f yore-(«19d Rwracmiure C(C‘V’WACUV‘U (Shir: V3) 0: Todanac A 1 Jokusou.‘\/]QHL\€7 #L i 1 --——-- —- ----— fi— —- .--.- ~——--—-——- - — -——-—q '02 .05 J0 .10 '50 lu+€fcep+ of FM: (V‘LV ~ Bu“! TbtfrJ/H: RI; (owm-rwxloq) Figure IV-5 Experimental temperature dependence of thin wire resisti- vity ( cf. p. 26 for further explanation) {x‘ocnw't Had" wire (0) Morth‘cal 51_ ;‘ 'huwvwEML ‘ 3 (o) S E . . ,l-rw- {5 Cxplnwlp V but/h (A) if ’S '. Tammi-{mi 3v {5; O / Euchw ZULE’T. . '9') :1 (a) ,, Ftu+ctzn ‘ 0—" “Point 8 g, " .2 a» .l b °K TQM ()e rod‘urt Figure IV-6 Comparison of experimental wire and bulk resistivity with various theoretical resistivity effects (matched at 3°K. except for bulk values) for a particular wire (In 4) V. CONCLUSION The general nature of our results indicates, as should perhaps be expected, that the simple model used to investi- gate the various~mechanisms giving rise to resistivity in thin wires provides a picture which shows qualitatively fair agreement with experimental results, but which is in- sufficient to give correct or even conclusive quantitative answers. Many difficulties arising in a Fuchs type treat- ment have been avoided only by using the crudest physical and mathematical approach possible; any great elaboration would re-introduce these same problems also in this type of kinetic approach. It has been shown that a mechanism such as suggested by Olsen can account, at least to some extent, for the temperature dependence of the resistivity of thin wires, and, further, that effects due to Umklapp-processes and e1ectron-electron-to-surface mechanisms should at any rate not be excluded a priori in a more rigorous future in- vestigation. Before the relative importance of these me- chanisms can be evaluated, however, further investigations on bulk phenomena (such as a determination of the contribu- tion of Umklapp-processes to the total bulk resistivity, or the calculation of electron-electron mean free paths in bulk materials) would have to be undertaken. Even though it would seem that in the samples investi- gated by Olsen electron-electron scattering is relatively unimportant, the arguments concerning the relative impor- tance of such effects in thin wires and in the bulk are I‘W ». I? .I. ‘4 -52- presumably correct. It follows that suitable measurements in thin wires may provide information on a mechanism, which, although it exists, cannot be observed in bulk materials. -55... VI. APPENDIX I. Derivation of the "distance of flight" for electronepho- non scattering. If (1(6)) is the "distance", NO) the "time" of flight between two collisions, then cue) : (rtce) where 6) is the scattering angle. Now 7 M W‘ 8’...— (“(039) SiquQ U9) I _ Pb nel E ne" and if we assume [(9) = mus-Lewd T," {or 059999.400: 00 {Or 9>ewm and éSwwu at! then 9ma< H I 3 MA | <3 ’2'— "'" :\ 6/ M9 3: “(1". 9:.” T ne" L 2. HC’ c. 8 Hence '4. 1'7) But we have and thus obtain , { WT)” } .L = wt. = “*“‘ 5 Q [Zne‘mwfc‘ ('31- which is the cited result. -34- II. Outline of the multiple scattering probability deriva- tion. (1) Single Event. Here we know the probability per unit time that an elec- tron is scattered, by a single collision, through an angle 9 into a solid angle do ?t(9) r190 ‘ (plate ”P691 3 U70. dud {dz diHauu 0,; “(3H O s Gé ewux Then "171(6) duo :: 6‘ 3(9) Jun) where 3(9) is the corresponding probability per unit length traversed. Hence l \ .. | 3(9) = F 12(9) "' ‘6; ) Then the mean square scattered angle for an electron which C) $.59££ GQMAQX has traversed a small distance ox is ewax <6 >4»: ox j 913(6)6d9-ZW = 3:1 AX o H where '/ Lia/7T 914M] Z 00' (2) Multiple Scattering. We cannot use here immediately'fl(6), which was the transition probability for the single event. Instead we use a multiple scattering transition probability(gflvdefined SHCh that 00 (mm) we 3 1 ~00 co and jelfiute) AG = (9‘)!” = I?" AK _00 . where the limits are extended since RmJG)is assumed to have a sharp maximum about 6 r O and then go to zero. The (3A,,(9) thus defined gives the probability of an electron being scattered through an angle 65 by mgpy scatterings (each of which is through a small angle) after having traversed a distance AnQ along the direction of initial motion (the x-axis). If the angle of "multiple" scattering or the tra- versed distance Ax. is made very small, the problem should reduce to the single scattering event, hence the second of the above requirements. Using this probability, one findsqa) the probability 665,9) that an electron, after traversing a distance 3? , is deflected by an angle 9' from its initial direction, and further the probability H(3<',s) that an electron, again after going a distance i' along the wire axis, is deflected a distance 8 perpendicular to the wire axis. These two probabilities are given by (7(7‘9) = 2%"2 €XP [Ell—r”; 9770”} -' ‘ LOUISW _ S 2. 1 ~ '3 H