Seebeck effect of the paramagnetic polaron

Solid State Communications, Vol. 9, pp. 2111—2114, 1971. Pergamon Press.

Printed in Great Britain

SEEBECK EFFECT OF THE PARAMAGNETIC POLARON Akira Yanase* Ruhr Universit~t Bochum, Institut für Fxperimentalphysik IV, Bochum, West ~rmany

(Received 15 April 1971 by P.G. de Gennes)

The thermoelectric power, TEP, of the paramagnetic polaron is studied theoretically. It is shown that the entropy of the polarized spins in the polaron state gives an anomalous TEP, which has the opposite sign to the usual TEP, and becomes infinitely large in the low carrier density limit at the Curie temperature. As a consequence of this theory, a unique behavior of the TEP in n-type CdCr 2 Se4 is explained, where a minimum of the TEP coincides with a maximum of the resistivity. 35 Moreover, the above the temperature. mobility ofCurie the magnetic polaron was also discussed.6 The purpose of this note is to propose a theory of the TEP of the magnetic polaron. As a consequence of this theory, a straight-

THE TRANSPORT properties of magnetic semiconductors have been extensively studied recently. Among others, Amith and Gunsalus reported measurements of the Seebeck coefficient (thermoelectric power) in a series of samples of Cd 1_~In~Cr2Se4between 4.2°Kand 3000K.l The critical feature in the data is that a minimum at about 150°Kin the thermoelectric power, TEP, coincides with a maximum in the resistivity, which is near, but above the Curie temperature of about 130°K. These phenomena cannot be explained clearly in terms of the usual simple theory of semiconductors. Usually the decrease of the absolute value of the TEPand indicates theincrease decrease of the chemical potential hence the

forward explanation of the observed experimental results for Cd,_~In~Cr2 Se4 can be demonstrated. Let Q be the absolute thermoelectric power of any substance and 77 be its absolute Peltier coefficient, defined as the heat flux carried to the terminal by a unit current. At any absolute temperature7 T these quantities are known to obey the relation Q=7T/T. (1)

of the carrier density. Friedman and Amith have tried to explain these phenomena assuming an unusual temperature dependence of the positions of the band edges of a rather complicated band structure.2 However, at present there is no direct experimental evidence which supports this assumption.

As Wagner has pointed out, equation (1) provides an easy way of calculating the TEP of any model of a metal or semiconductor for which the problem of the isothermal electric conduction is solved: one merely need calculate i~ from this solution.8 Assuming various kinds of conduction states and following the definition mentioned above, 77 is given by

In recent works, we have shown that the paramagnetic polaron will be present in these materials *

77

Permanent address: Department Physics,

=

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(2)

e 3 n~j~

Faculty of Science, Tohoku University, Sendai, Japan.

where n, and p~ are, respectively, the number 2111

2112

SEEBECK EFFECT OF THE PARAMAGNETIC POLARON

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Vol. 9, No. 23

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FIG. 1. The spin entropy shown as a function of temperature. The unit of entropy is Boitzmann’s constant, and the unit of temperature is the Curie temperature. The number beside the line gives the value of 100 (kO/IS), where 0 is the Curie temperature, I the s—d interaction constant and S the spin value of the localized spin, here S = 3/2. The value for IS/E 5, where Eb is the band width, is given in the figure with respective marks in the graph. The corresponding value for the TEP can be5 seen on the left scale in mV/deg. All states shown in the figure are those of the large magnetic polaron. and mobility of the carriers in the j-th conduction state, and e 3 is — el for electrons and + el for positive holes, and e the electron charge. Moreover a1 is the partial entropy defined as the entropy change when one carrier is added to the j-th state and is given by =

a~/a~2,

(3)

where a is the total entropy of the system. Note that the suffix j in equation (2) indicates not only the band suffix but also the wave vector in the case of usual band conduction. Moreover, j should run over the phonon modes in the numerator, if necessary (phonon drag case). Therefore one should use an energy dependent p~,that is, the correct solution of the isothermal electric conduction.

In usual semiconductors the entropy carried by the carriers is only the population entropy, that is, the entropy due to the excitation of the carriers to the conduction state, which is always positive. However the magnetic polaron is accompanied by highly polarized spin moments nearby. These polarized spins of course have a smaller entropy than the non-polarized ones; therefore we have a positive additional term of the TEP for the magnetic polaron of the electron. The carrier number n2 should be determined by minimizing the free energy, which is given by

F

=

n3 (F~

[N~ ln N~ — n3 ln nj

—

(N2

~)

—

—

÷kT £~

n,) in (N,

—

n3)], (4)

Vol. 9, No. 23

SEEBECK EFFECT OF THE PARAMAGNETIC POLARON

where F, is the free energy of the state j and N, is the total number of the j-th state and t~ the chemical potential. The result is that N~ =

1

+

exp [(FJ

polarons overlap with each other. It should be

2

magnetic semiconductor will become the magnetic impurity state, MIS, and that the interaction between the polaron and MIS has the same effect, since the localized spins around the MIS is highly

t~)/kT1

This expression is not the usual one, since F

contains the free energy of the polarized spins if the state is the magnetic polaron state, whereas in the usual formula F, could be the level position of the j-th state. Therefore, if we can assume that only the magnetic polaron can contribute to the electric conduction, the TEP is given by

Q

=

—

[ok, + (F~

—

remarked in reference 5, when the concentration of polarons becomes large, the interaction between the polarons supresses this divergence, because the spin polarizations of neighbouring

(5)

. —

2113

~)/TI lel,

(6)

where F~ is the free energy of the magnetic polaron and a~, is the entropy of the polarized spins nearby the polaron state, and is always negative. The basic equation for the paramagnetic polaron is given by 2I + ~m~nm <5~~>l, (7) = G[g1LL~H0 + C~! — Im Enm Cm — 1 = EC~ , (8)

c~

where I is the s—d exchange constant, the spin polarization of the localized spin, C,~the amplitude of the self-trapped electron and G (X) the usual formula for the magnetization. As shown in reference 3, the region where the spin polarization is almost saturated, extends over many lattice sites and increases when the temperature decreases toward the Curie temperature. This implies that the absolute value of c~, is large and increases with decreasing temperature. The accurate value for a~ can be calculated from the numerical solution of equations (7) and (8) which has been given in reference 5. Examples of the calculated results are shown in Fig. 1. The absolute value of the entropy increases to infinity near the Curie temperature and the critical exponent is given by 1/2. Therefore, if an ideal situation of low carrier density in a very pure magnetic semiconductor can be achieved, the TEP must become positively infinite for electron conduction. Note that this sign is opposite to the usual TEP. However, as

noticed here that the impurity state in the

polarized. Moreover if different kinds of carriers start to contribute to the electrical conduction, the anomalous TEP is again suppressed as seen from equation (2). As a consequence of this theory, we can give an explanation of the experimental results for CdCr2Se4, as follows. At high temperature, the TEP is given by equation (6), and the second term is calculated from the activation energy for the Hall coefficient. From experiment, this value is about 0.3 eV, and gives a TEP of —1 mV/deg and —2mV/deg for 300°Kand 150°K, respectively. As shown in Fig. 1, the first term increases rapidly decreasing and consequently thewith absolute value temperature, of the TEP decreases. Taking 100 kU/IS = 2.0 in Fig. 1, the total TEP is —0.5 mV/deg and 0.2 mV/deg for T = 300°K and 150°K, respectively, in good agreement with experiment.’ In these substances, the s—d exchange is rather large as indicated by the red shift of the absorption edge’° and determined from the TEP here. Therefore the s—d exchange potential has the same order of magnitude as the impurity potential. When the temperature decreases toward the Curie temperature, the region of the polarization of the localized spins around the MIS becomes larger. Then the radius of the MIS must increase. As this takes place simultaneously, the radius of MIS changes rapidly around the Curie temperature. If the radius of the MIS becomes larger than a certain critical radius, which is determined by the density of the MIS, the substance becomes a degenerate semiconductor as observed in highly doped semiconductor.9 Experimentally, a rapid decrease of resistivity starts at 150°K. Therefore we expect the radius

2114

SEEBECK EFFECT OF THE PARAMAGNETIC POLARON

of the MIS to reach the critical value near this temperature. Below this temperature, the carrier density increases rapidly suppressing the anomalous term of the TEP due to spin polarization by the mechanism mentioned above. In this way, the minimum in the absolute value of the TEP can coincide with the maximum of the resistivity, When an external magnetic field is applied, the minimum in the TEP shifts to a higher temperature indicating that the temperature of the transition to a degenerate semiconductor increases. In order

Vol. 9, No. 23

to get a more quantitative agreement with experiment, we should consider more quantitatively the effects of the interactions between the magnetic polaron themselves and the interaction between the MIS and the magnetic polaron. The latter will be especially important in the determination of the concentration dependence of the TEP.

Acknowledgements

—

express his thanks to discussions.

The author wishes to J. Kiibler for valuable

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YANASE A., to be published in tnt. J. Magnetism.

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KASUYA T., YANASE A. and TAKEDA T., Solid State Commun. 8, 1551 (1970).

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DE GROOT S.R., Thermodynamics of Irreversible Processes. Interscience, New York (1952).

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WAGNER C., Z. phys. Chem. B22, 195 (1933).

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KASUYA T., to appear in Proc. Rev, mod. Phys. 40, 677 (1968).

10.

82,

434 (1970); FRIEDMAN L. and AMITH A., IBM J.

mt. Conf. Semiconductor,

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BUSCA G., MAGYAR B. and WACHTER P., Solid State Commun. 7, 965 (1969).

Die termoelektorische Kraft, TEK, des paramagnetischen Polarons wird theoretisch untersucht. Es wird gezeigt, daB die Spinentropy, die zu dem magnetischen Polaron gehort, eine anomale TEK ergibt. Diese hat die gegenteilige Wirkung von der gewohnlichen TEK und wird unendlich groB in Grenzfall kleiner Ladungsträger-Konzentrationen für Temperaturen nahe der Curietemperatur. Damit wird eine ungewbhnliche TEK in Cd,_~In~Cr S2e4 erklärt.

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