R 455 hilis Res. Rets 17, 337-343, 1962 RELATIONS BETWEEN CURRENTS AND VOLTAGES IN STRUCTURES CONTAINING SEMICONDUCTORS Summary by F. H. STIELTJES A general exression is derived for structures artially consisting of semiconductors, relating currents and voltage dros with injection, i.e., non-equilibrium henomena, by introducing the concet of injectionor extraction-induced e.m.f.'s or counter e.m.f.'s. An analogy is drawn with thermo-electric henomena. Résumé On tire une exression d'ordre général our des structures comortant artiellement des semi-conducteurs, associant des intensités et des chutes de tension à des hénomènes d'injection, c'est-à-dire de déséquilibre, en alicant le concet de la f.e.m. ou de la force c.e.m. induite ar injection ou extraction. On établit une analogie avec les hénomènes therrnoélectriques. Zusammenfassung Es wird ein allgemeiner, für teilweise aus Halbleitern bestehende Gebilde gültiger Ausdruck für die Beziehung zwischen Strömen und Sannungen unter Nichtgleichgewichtsbedingungen abgeleitet, indem der Begriff einer durch Injektion- oder Extraktion induzierten EMK oder Gegen- EMK eingeführt wird. Die Analogie mit thermoelektrischen hänomenen wird gezeigt. 537.311.33 1. Introduetion It will be shown that voltage-current characteristics of most structures artially consisting of semiconductors may be described unambiguously by ostulating the resence ofinjection- or extraction-induced counter e.m.f. 's (or e.m.f.'s if the device is ower-roducing and not ower-consuming) in series with resistive voltage dros. The result has been anticiated by Reiss 1), but the author's derivation and final resentation is different with the aim of showing its much wider significanee and alicability. A recent book by J. Tauc 2) gives a hysical treatment of the roblems of ower-roducing devices which is strongly akin to that given in this article. 2. Derivation Les us assume two different mobile charge carriers (e.g., but not necessarily so, electrons and holes) for which we may write exressions for the individual current densities as follows: j [i= -al grad ~1, ~ [a = -a2 grad ~2, ~ (1)
338 F. H. STIELTJES in which the j's are current densities, the e'sare quasi-fermi otentials exressed as voltages for both current carriers as defined by Shockleyê), i.e. 1>M =!J.M*/qM;!J.M* is the total electro-chemical otential er carrier of tye m and qs«is the charge of the same carrier. (Brenstedt 4) would call a otentialof the kind of 1> the electro-motive otentialof the electronic secies under consideration.) As is well known,!j.m* = qmt +!J.M* chem in which t is a sort of coarse electrostatical otential and!j.m* chem is the "urely" chemical art of the total thermodynamical otential, 'consisting of a concentration-deendent contribution as well as a contribution due to the influence of the surrounding medium; so em = t +!J.M* chem/qm. Finally the a's are scalar secific conductivities mostly written as!qm!!j.m CM,!J.M now 'being a mobility, CM the concentration. In a general discussion it is aroriate to use a without further secification. ' If one sums the two equations (1) and divides by UI + a2 one gets j al --- = - --- grad ei a1+a2 a1+a2 or rewritten, grad em = - j + A grad ed, (20) with j total current density, = 1 al + 0"2 ; further by definition: In the differential equation (2a) one recognizes already the resistive voltage dro j dr and the injection-controlled e.m.f. A grad ed dr because ed would be identically zero without injection (or extraction). Often it is worth while integrating eq. (2a) between two oints and ; one then gets em- em = I oj dr + fa grad 1>D dr, (3) em - em being the total mean voltage difference aearing or alied between oints and. It is obvious that besides being well-defined, j and ed have an unambiguous and clear hysical meaning, j being the divergencefree total current density and 1>D being a measure of the deviation from equilibrium at any oint due to abnormal oulations of. the two electron bands only, not being deendent on the "electrostatic" otential t at the oint considered. The imlications of these facts will be dealt with more thoroughly in the next section. On the other hand, em does not share the roerty ofhaving
. RELATIONS BETWEEN CURRENTS AND VOLTAGES IN SEMICONDUCTORS 339 a clear-cut hysical meaning although it still is well-defined. The discussion of this oint will be treated after the next ste, which is erforming a artial integration on the last terms. One then gets CM - CM = f j dr.; [(A~D) - (A~D)] + f ~n grad A dr. (4) If one now chooses the oints and in regions where ~D vanishes identically the second term at the right dros out and one gets ~M - CM = f j dr + f cn grad A dr. Equations (3) and (4a), which are equivalent, given the just-mentioned condition for the oints and, differ in aearance. Equation (3) seems to suggest that injection-induced e.m.f.'s are resent where there is a gradient in ~D, A then being a measure of weighting these influences, whereas eq. (4) seems to imly that the seats of the induced e.m.f.'s are at the laces where A shows a gradient, ~D now taking the role of a weighting factor. About the ambiguity of ~M the following may be remarked: CM is a sort of weighted mean of the two c's, CI and C2. One might just as well have taken another weighted mean ~M', e.g. (4a) but still Then CI = CM' + ta cn + HD, C2= ~M' + tacn - HD, in which a may be a variable in the coordinates. erforming that led from eq. (1) to eq. (4a) one gets the same oerations,(cm' + ta~d)- (CM' + 1- á~d) = f o j dr + f ~D grad A dr. (4b) Obviously the ambiguity in choosing the right weighted mean is not imortant under the conditions imosed on the oints and. Then all mean values coincide, and become the electromotive otentials at these oints. This includes oints in materials like metals where there is no sense in seaking of different tyes of electrons. For these materials one cannot write two equations (1) but only one for the total current j = -(j grad c; ~ may be called ~M or CM' as one wishes. Only at a junction between a metal and a semiconductor will the two ~'s, ~I and ~2, start to diverge. Exression (4a) is anticiated by Reiss' article. The difference in aearance will be discussed in the next section.
340 F. H. STIELTJES 3. Discussion of the result Exressions (3) and (4) are very generally valid, e.g., it is not necessary to secify how the distribution of cd has arisen *). These exressions 'reflect the fact that the hysical state at any oint (including the local energy) needs an extra determining quantity for its comlete descrition; with other words, there exists a new degree of freedom due to the ossibility of having oulations of the electrons in the different energy bands not in equilibrium with each other. The last terms in the exressions (2a), (3) and (4a) are connected with a transfer of "normal" energy to the new non-equilibrium energy or vice versa. In a way these terms act as corrections to the normal interlay of ower in a conductor described by j and cm only. What haens with the extra energy, or ower in the way of migration, suly (e.g., by illumination), disaearance (e.g., by roduction of light or by radiationless recombination) is of no concern in this stage. Of course for ractical alications in rincile the fate of this extra energy does matter, as will be discussed in the next section, but the exressions (2) to (4) still have the merit of their generality and of their hysical content sketched above. In view of the receding discussions it is not surrising that there is a strong analogy with the thermo-electric case. There we find formulas like grad c = - j + a grad T, c - c = f j dr - f a grad T dr, c - c = f j dr + ft grad a dr, (2') (3') (4') in which c = electromotive otential, T = temerature and a = thermo-electric coefficient. The ste from (3') to (4') needs also an extra condition: and should have the same temerature. The analogy can be worked out quite nicely. In the thermo-electric case one also needs an extra arameter, viz. the temerature, to define the hysical state roerly and there is also a ossibility of transfer of electrical into thermal energy and vice versa. Finally it should be remarked that one might relace cd by a local artial interband temerature, which governs the distribution over the different bands, as is done when only a few discrete levels are available for electrons. However, in cases of discrete b~nds of very many levels this temerature needs a very careful definition. *) Reiss, not introducing <D, gives an exression deending on recombination rate and divergence of electron (or hole) current equal to <D under the secial assumtions he studied.
RELATIONS BETWEEN CURRENTS AND VOLTAGES IN SEMICONDUCTORS 341 4. ractical alications In general equations like (3a) or (4a) are not sufficient to solve a roblem of current-voltage characteristics. The distribution of cn within the structure has to be known too. This means solving the roblem of the distribution of concentration of charge carriers, given equations (1) as well as the generationrecombination equations, which are generalized continuity equations. This is quite a formidable job, certainly so if the geometry of the structure is comlicated. Even then the alication of formulas like (3a) and (4a) are very helful, because the calculations of cn and o from the concentration is veryeasy, and the rest, the integration along a chosen ath, is in rincile straightforward. Certainly it is less troublesome than the method mostly used of calculating extra electrostatic otential jums at junctions, sulemented by calculations of Dember-otential differences. In cases of simle geometry, leading to a one-dimensional roblem, e.g. (quasi-) lanar, cylindrical of shericallayer structures, the rofit of the exressions becomes even greater. The solution of the roblem of the distribution of Cn becomes much easier to begin with, but now even a quick insection may give useful results. Very often such structures consist of layers of different materials (- or N-tye semiconductors, metals) homogeneous in each art with abrut junctions. To a first aroximation the A's in each art are constants *); so eq. (4a) is very aroriate for this case. because grad A equals zero everywhere but for the junctions, where A is a ste function and grad A is a delta function. This means one has only to know the value of CD at the junction, or, in reverse, it offers the ossibility to calculate quickly a CD aearing at a junction if one alies an external voltage to a structure of this kind. This resembles strongly the case of a thermo-electric device in which it is assumed that the thermo-electric coefficient a is only deendent on the material; then knowledge of temeratures at the junctions only is already sufficient if the two materials are homogeneous in themselves. The analogy even carries on; quite often in the thermo-electric case there is a slight deendence of a on the temerature (the so-called Thomson-heat is connected with this) and it is necessary to know the temeratures within the material too to be able to calculate this Thomson-effect correction. In the case of injection-induced e.m.f.'s or voltage-induced injections, it is necessary to correct for the variation in A due to the injection, i.e. the CD at each oint. *) If.l refers to holes and.2 to electrons, A equals very recisely --!- for ronounced -tye and +-!- for ronounecd N-tye material, whereas for intrinsic material without traing it 1 f.l. - f.lh ~ f.l. mobility for electrons, equais T ---. bili ti h I f.l. + f.l1' f.lh mo I ity or 0 es.
342 F. H. STIELTJES 5. Limits of alicability and ossible extension The results are derived by urely analytical means from eqs (1); so limits of alicability have to do with hysicallimits inherent in these equations. We shall insect a few of these.. (a) Because no secification of a is needed it is absolutely ermissible to imly abnormal deendence of a on concentration for instance (so long as it stays a scalar quantity). It would even be ermitted to assume a grad e deendence (a fieldstrength deendence is well known) for a, but for the fact that this would robably turn a scalar a into a tensorial one even for isotroic materials. (b) Equations (1) might be too simle, due to "drag" effects, and should be relaced by Ii = -au grad ei - ai2 grad e2, Ï2 = -a21 grad ei - a22 grad e2. Normally ai2 = a2i (Onsager relation) but this has not to be assumed at all. If one goes one ste further one will see that this extension is quite ermissible, because summing results into j = -(au +ai2+a2i +a22) grad em -!(au-ai2+a2i-a22) grad ed. So we have only to redefine 1 - = au + ai2 + a2i + a22 and 1 au - ai2 + a2i - a22 A=- 2 au + al2 + a2i + a22 (c) It may haen that ei and/or e2 are no longer definable. This occurs in a junction biassed in the reverse direction carrying current vacuum-diode-wise. Then it is always ossible to find regions outside the barrier but very close to it where the non-thermal velocity distributions of the electrons and holes have died out, so that e's may again be defined. The current densities being known there too it will always be ossible to substitute the region where it went wrong with the c'sbya well-behaved region having the same transfer characteristics. (d) The materials are no longer isotroic (or cubic) so the secific conductivites as well as the A's are no longer scalars but tensors, which need not be symmetric in addition (e.g., due to the resence ofmagnetic fields). In the general case the alication of equations like (2) and (3) with tensorial coefficients becomes very comlicated, even in the case of simle geometries and homogeneous materials (with and without abrut junctions). The ste from eq. (3) to eq. (4) becomes in general rohibitively unrewarding, because artial integration is only ossible after secification of the ath of integration, the result being deendent on this ath. There is an imortant excetion where things become simle again, viz.
RELATIONS BETWEEN CURRENTS AND VOLTAGES IN SEMICONDUCTORS 343 when current-density vectors and gradient vectors still coincide everywhere. This is the case with quasi-lanar structures with layers always erendicular to one of the rincial directions of all the symmetric tensors. This imlies that all tensors have at least one coinciding rincial direction, which in true for many crystal classes. Then all derivations still hold if one substitutes for the q,'s and A the values say Uzz and A zz they have for that referred direction erendicular to the layers. (é) A few final remarks: (1) It is not necessary that 1>1 and 1>2 refer to electrons and holes (it might also be e.g. light and heavy holes). (2) It is ossible to extend the ideas followed in this article to cases where there are more than two tyes of mobile charge carriers, or where thermo-electric effects as well as injections are taken into account, or even where one considers also the transfer of electrical and normal chemical energy like in batteries. Eindhoven, June 1962, REFERENCES 1) H. Reiss, J. al. hys. 27, 530-537, 1956. 2) J. Ta uc, hoto and thermoelectric effects in semiconductors, ergamon ress, Oxford 1962. 8) W. Shockley, Elektrons and holes in semiconductors, D. V. Nostrand, New York, 1950,.308. 4) J. N. Brönstedt, hysical chemistry, W. Heinemann, London, 1937, cha. IX 2.