The theory of the galvanomagnetic and thermomagnetic effects in metals By E. H. Sondheimer, Trinity College, University of Cambridge ( Communicated by A. II. Wilson, F.R.S.—Received 27 August 1947— Revised 8 January 1948) The methods of a previous paper are used to discuss the effect of a magnetic field on the thermoelectric power of a metal containing two overlapping energy bands of normal form. Exact solutions of the transport equation are obtained for the three limiting cases of high temperatures, low temperatures and very strong magnetic fields, and it is shown that the formulae can be generalized to give approximate expressions for all temperatures and all fields. The magnetic change of the thermoelectric power is found to be small at very low and high temperatures, and to pass through a maximum at intermediate temperatures.
Galvanomagnetic and thermomagnetic effects in metals 485 by certain coefficients which may be defined by considering a metal strip with its surface in the ary-plane, carrying an electric current Jx or a thermal current wx in the ^-direction and subjected to a magnetic field H in the 2-direction. The following cases are then of practical importance: If J#x= 0 but the transverse current Jy = 0, there exists a transverse electric field Sy. The ratio $y\HJx is known as the Hall coefficient AH.
486 E. H. Sondheimer particularly at low temperatures where the effect is most pronounced but where the theory is more difficult owing to the non-existence of a time of relaxation. It is not necessary to consider the remaining thermoelectric effects, since they can all be expressed in terms of the thermoelectric power (see, for example, Wilson 1936, §§5*41 to 5*43). Also, as in I, the case of a longitudinal magnetic field has to be excluded, since the two-band model is not sufficiently general to give a non-zero result for this case.
Galvanomagnetic and thermomagnetic effects in metals 487 upon r and the first derivatives of E; and in fact only the squares of the latter occur. It is therefore unnecessary, if one restricts oneself to order of magnitude calculations, to consider a model in which the first derivatives change sign. This is the justification for the very wide use of the free electron model for discussing the electrical conduc tivity of metals whether monovalent or not. In some ways it is better to use the two-band model for divalent metals since the meaning of the parameters becomes clearer, and it is possible to see whether, for example, the average effective mass of the electrons is the arithmetic or harmonic mean of the effective masses of the in dividual electrons. If, however, the parameters in the more general models cannot be distinguished by experiment from those for simpler models, it is best to choose the simplest model that gives a non-zero result.
488 E. H. Sondheimer unnecessary in the present state of development where the experimental data are so inadequate.
Galvanomagnetic and thermomagnetic effects in metals 48H The change of the thermoelectric power in a magnetic field at various temperatures has been investigated experimentally by Griineisen & Erfling (1939) for beryllium. The effect is found to be small at high and very low temperatures and to pass through a maximum value at an intermediate temperature, the position of the maximum moving towards higher temperatures as the magnetic field is increased. It is shown in § 4-21 that .the predictions of the two-band model are in general agreement with this behaviour.
490 E. H. Sondheimer 1935); it is very seldom that the various effects have been measured on the same specimen and no attention seems to have been paid to choosing metals for which the theoretical* interpretation would be relatively easy. The inconsistency of the results available is shown in table 1 infra, where data are given for a quantity which should by thermodynamic arguments be unity but which differs very considerably from it. If we disregard ferromagnetic metals, strongly anisotropic metals such as bismuth and discard results which differ by a factor of more than 2 when measured by two different experimenters, there is very little experimental evidence left. Further, there are no data at all for those metals in which the parameters for the separate s- and d-bands can be estimated from other phenomena, so that it is only possible at present to attempt to verify the theory as regards orders of magnitude, and even then only by applying the principles laid down in § 1*2 supra and in I, p. 448 for interpreting the parameters.
Galvanomagnetic and thermomagnetic effect4s 9i1n metals — ebeing the electronic charge and £ the Fermi energy level, then — <3/e is known as the absolute thermoelectric power per degree. It will be assumed, as in the case of the thermal conductivity, that (3 is measured under adiabatic conditions, the thermal conditions in the ?/-direction being such that wy = 0. Using this condition in addition to Jx = Jy = 0, it is found by means of I, equations (13) to (18), that (P. - Pi) +vd) (x ,+Xd) - (W.- Wd) (rs- r d)} _ ________________+ + (lg ~ Id) + ~ (^s + ^d)} m (S. 1.2m (Qs Qd) {(Vs (Ws {(Vs + Vd)*+(Ws-Wd)*}(Xs + Xd) ’ where k is Boltzmann’s constant, V, W, X, aYre certain combinations of the i(n)’s defined by I, equations (20) and (22), while ?P = if- = ~ (2) and the suffixes s and d refer to the s- and d-band respectively.
492 E. H. Sondheimer the reader. It will accordingly be assumed that the value of the Ettingshausen- Nernst coefficient which is actually measured is obtained by using the condition 3 Tjdy = 0 instead of wy = 0. Denoting this value by B^N, it is found that Ri k*T(Qs+ Qd) (Vs+Vd)- (P8-P d) (Ws-Wd) E N ~ eH(V8+Vd)* + (W8-Wd)* ' (b) Finally, for the Righi-Leduc effect dT/dx^O, while Jx — 0, wy = 0, and 1 Y*-Yd Brl HXs + Xd‘ (7) The measured values have again to be corrected for any heat which is lost to the surroundings.
Galvanomagnetic and thermomag effects 4in93 metals 4. The thermoelectric power 4* 1. Single-band model. At high temperatures a time of relaxation exists, and the thermoelectric power in a magnetic field of arbitrary magnitude can readily be calculated by means of equation (1) together with (8) and I (27), (28). The resulting expression is complicated, and owing to its restricted validity it will not be considered further here. It is of greater interest to consider the possibility of setting up an interpolation formula, analogous to the expressions obtained in I, which can be used for all temperatures and magnetic fields. By analogy with the procedure in I, §§ 4 and 6, such a formula would be expected to depend in some way on an inter polation formula for the thermoelectric power of a metal containing a single band, corresponding to I, equations (54) and (73) for the electrical and thermal conduc tivities. No such formula has so far been given, and it is therefore necessary to begin by supplying it here. The formula is obtained by writing down the exact expressions for © in the three limiting cases where the solution is known, and guessing a generalized expression which reduces to the correct form in these limiting cases.