Spin fluctuations and properties of the thermoelectric power of nearly ferromagnetic iron monosilicide. A. G. Volkov, A. A. Povzner, V. V. Kryuk, and P. V. ...
PHYSICS OF THE SOLID STATE
VOLUME 41, NUMBER 6
JUNE 1999
Spin fluctuations and properties of the thermoelectric power of nearly ferromagnetic iron monosilicide A. G. Volkov, A. A. Povzner, V. V. Kryuk, and P. V. Bayankin Ural State Technical University, 620002 Yekaterinburg, Russia
~Submitted August 20, 1998! Fiz. Tverd. Tela ~St. Petersburg! 41, 1054–1056 ~June 1999! The thermoelectric power of nearly ferromagnetic iron monosilicide, which passes through an electronic semiconductor–metal transition with increasing temperature, is investigated theoretically. The results of this investigation indicate that a sizable paramagnon-related increase in charge carriers can occur in nearly ferromagnetic semiconductors, and that spin fluctuations can modify the electronic spectrum and thereby renormalize the diffusion component of the thermoelectric power. The transition from semiconductor to metal decreases the paramagnon component sharply and the thermoelectric power changes sign, which agrees with experimental data for iron monosilicide. © 1999 American Institute of Physics. @S1063-7834~99!02706-9#
1. Iron monosilicide ~FeSi! is typical of a class of nearly ferromagnetic semiconductors that undergo a semiconductor–metal transition with increasing temperature. This transition is accompanied by disappearance of the energy gap in the itinerant d-electron spectrum between the ‘‘valence’’ and ‘‘conduction’’ bands.1,2 It also causes a sharp increase in the magnetic susceptibility x (T), 3 and, as experiments on inelastic neutron scattering4 show, a considerable increase in the amplitude of spin fluctuations in the d-electron system. According to spin-fluctuation theory, in nearly ferromagnetic materials the d-electron energies are spin-split in the fluctuating exchange fields j, leading to renormalization of their density of states5,6 g ~ «, j ! 5
( g ~ «1 s 8 j ! /2.
conductivity.10 This latter data reveals the dynamic nature of the spin fluctuations, along with the existence of a characteristic time t s f ;10212210213s, 11 and also the fact that the d electrons are the majority carriers of electric current.10 However, the question of how these dynamic spin fluctuations affect the thermoelectric power of nearly ferromagnetic FeSi, which should be most sensitive not only to spin fluctuation excitations but also to features of the transformation of the electronic spectrum, has remained unaddressed up to now. 2. In this paper we will discuss how spin fluctuations affect the temperature dependence of the thermoelectric power in almost ferromagnetic semiconductors ~among them iron monosilicide! using the theory developed in Refs. 5, 6, and 8–10. In calculating the diffusion component of the thermoelectric power of d electrons we will use the well-known relation from Ref. 12, while incorporating the spinfluctuation-induced renormalization of the electron spectrum and the density of d-states ~see Eqs. ~1! and ~2!!:
~1!
In almost ferromagnetic semiconductors this process also changes the width of the energy gap between the valence d-band and the conduction d-band: E g ~ j ! 5E g ~ 0 ! 22 j ,
S d ~ T ! 5I 1 ~ j ! /eTI 0 ~ j ! ,
~2!
where the kinetic integrals are
which eventually disappears due to the monotonic increase in j (T). Here s 8 561 is the spin quantum number corresponding to axes of quantization connected with the spatial and temporal fluctuations of the j fields, g(«) is the density of states of noninteracting d-electrons, j 5Q A^ m 2 & , ^ m 2 & 1/2 is the amplitude of spin fluctuations, Q is the parameter for intra-atomic Coulomb interactions, and E g (0) is the width of the forbidden band in the spectrum of noninteracting d electrons. The hypothesis that spin-fluctuation-induced renormalization of the d-electron spectrum occurs in FeSi is indirectly confirmed when the results of calculations are compared with experimental data not only for the magnetic susceptibility5 but also for the heat capacity,7 the temperature coefficient of thermal broadening,8,9 and the electrical 1063-7834/99/41(6)/3/$15.00
~3!
I n~ j ! 5
E
`
2`
S
w ~ «, j !~ «2 m ! n 2
w ~ «, j ! 5k 2 ~ «, j ! t
D
] f F ~ «, m ! d«, ]«
d« , dk
~4!
~5!
e is the electron charge, f («, m ) is the Fermi–Dirac function, the electronic quasimomentum k(«, j ) is defined via the function g(«, j ) in the effective-mass approximation, and the relaxation time
t 5Ck 2r21
d« dk
for r53/2 corresponds to scattering by phonons. 960
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Volkov et al.
Phys. Solid State 41 (6), June 1999
The function g(«, j ) is modeled with the help of Eq. ~1! in conformity with the results of band calculations and the expressions for the spin-fluctuation amplitudes given in Refs. 6, 7, and 9:
j 2 5Q 2 m 2 5Q
(q
E
`
0
f B ~ v /T ! Im~ D 21 ~ j ! 1X ~ q, v !! d v
'Q 2 B 2 D ~ j !~ D 21 ~ j ! 1a ! 21 . ˜ ~ m ! /3! D ~ j ! 5 ~ 122Qn e f ~ j ! /3j 2Qg
21
~7!
is the exchange enhancement factor of the magnetic susceptibility,6,7 ˜g ~ m ! 5
P s 8 561 g ~ m 1 s 8 j ! , g~ m,j !
ne f ~ j !5
1 s8 2 s8
(
E
`
0
f F ~ «, m ! g ~ «1 s 8 j ! d«
~8!
is the effective number of magnetic carriers, X ~ q, v ! 5Q„x 0 ~ 0,0! 2 x 0 ~ q, v ! …5aq 2 2iB v / ~ qQ ! . Here q is the quasimomentum vector in units of the magnitude of the Brillouin vector, v is the fluctuation frequency, and x 0 (q, v ) is the Pauli susceptibility. The coefficients a and B, which specify the dependence of the susceptibility on frequency and quasimomentum, are determined either from band calculations or from the results of magnetic neutron scattering.4,11 For FeSi, according to Refs. 1, 7, and 9, we have a50, B56, and Q50.8 eV. Moreover, according to spin-fluctuation theory, the effect of paramagnon drag, which is analogous to phonon drag, should contribute to the temperature dependence of the thermoelectric power.12 In order to estimate this contribution, let us calculate the electron pressure caused by interaction of electrons with paramagnons,
] F pm , ]V
~9!
where V is the volume, and the paramagnon portion of the free energy is given according to Refs. 6, 7, and 9 by the expression F pm 5
F
5Q 2 1 Q„4g ~ m , j ! 2n e f / j … 3en 3 2D 21 ~ j !
Here
P pm 5
is the pressure, and V is the volume. A similar relation is obtained by using a crude approximation for the contribution to the thermoelectric power from phonon drag as well. Taking into account Eqs. ~6!–~8! and ~10!, we can rewrite Eq. ~11! in a form convenient for calculations as follows: S pm ~ T ! 5
~6!
(q
E
`
0
f B ~ v /T ! Im
D
21
X ~ q, v ! dv. ~ j ! 1X ~ q, v !
~10!
Then, assuming that the force caused by this pressure equals the force arising from the Coulomb interaction of the electrons due to their redistribution over the length of the conductor ~under steady-state conditions!, we find the following approximate expression for the paramagnon contribution to the thermoelectric power: S pm ~ T ! 5
1 ]P 1 ] 2 F pm 5 a , en ] V ] T en ] V pm
~11!
where n is the carrier concentration, which satisfies the relation n52n e f in the semiconducting phase, a pm is the paramagnon component of the thermal broadening coefficient, P
961
G
d ^ m 2& . dT
~12!
3. In order to compare these expressions for the thermoelectric power with experiment, let us use the results of calculations of the d-electron density of states of FeSi taken from Ref. 1, along with estimates of the spin-fluctuation amplitudes from Refs. 7 and 9. Then, using the function g(«, j ), we calculate the functions m~j! ~from the condition of electrical neutrality! and k(«, j ) ~in the effective-mass approximation!. Using Eqs. ~3!–~5!, we then can determine the diffusion component to the thermoelectric power of FeSi. In this case we find that that the diffusion component of the thermoelectric power gives a satisfactory description of the experimental data in the metallic region, but that it becomes much smaller than the observed values of S(T) in the semiconducting phase (T,T g '100 K). The latter fact indicates a need to include mechanisms for the drag of electrons by phonons or paramagnons. At this time it is not possible to estimate the effects of phonon drag in FeSi, due to a lack of information about the phonon spectrum and magnitude of the electron–phonon interaction. On the other hand, the value of the temperature that maximizes the phonon drag in the semiconducting phase, obtained by taking into account the temperature dependence of the concentration n and the phonon and paramagnon contributions to the coefficient a ~which, in conformity with the Gru¨neisen relation, is proportional to the corresponding contributions to the heat capacity!, is three times larger than that of the paramagnon contributions.~1! In the metal phase, the temperature at which the thermoelectric power connected with phonon drag is a maximum, according to Ref. 12, is estimated to be 0.15Q D ~for FeSi, Q D '560 K, see Ref. 13!, which does not correspond to temperatures at which FeSi is gapless.10 At the same time, the temperature at which the contribution to S(T) from paramagnon drag is a maximum ~50 K! is close to the experimentally observed value ~see Fig. 1!. In this case it is worth noting that the abrupt increase in concentration of mobile charge carriers n as the energy gap collapses ~see Eq. ~1!! can suppress contributions from the drag mechanisms, which according to Ref. 12 and Eq. ~12! are inversely proportional to n. Figure 1 shows calculated results for the temperature dependence of the thermoelectric power of FeSi, along with experimental data for a polycrystalline sample ~from Ref. 14!. Comparison of theory with experimental data shows that below T g the diffusion component is negligibly small and S(T)'S pm (T). In the gapless temperature range the contribution of the negative diffusion component is abruptly enhanced, so that the equation S(T)5S d (T)1S pm (T) was used in the calculations. In addition, dynamic spin fluctua-
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100 K. In this paper we do not discuss these alloys, since we feel that they could actually be two-phase systems, i.e., FeSi–FeSi2 ~Ref. 16!. Thus, our analysis of the thermoelectric power of FeSi shows that dynamic spin fluctuations can have a significant effect on the form of the function S(T) in nearly ferromagnetic semiconductors, by subjecting the current carriers to paramagnon drag and splitting the electronic terms. The mechanisms for paramagnon drag and spin fluctuation renormalization of the diffusion component of S(T) are important not only for nearly ferromagnetic systems, but also for transition metals in general and those of their compounds that exhibit itinerant ferromagnetism. This work was partially supported by a grant from the Competitive Center for Basic Science of the Ministry of General and Professional Education of the Russian Federation ~Project 95-0-7.2-165!. 1!
FIG. 1. Temperature dependence of the thermoelectric power of iron monosilicide: points—results of experiments in Ref. 14, solid curve—calculated results for the total thermoelectric power, which for T,T g coincides approximately with the paramagnetic component. The dots show the temperature dependence of the diffusion component of the thermoelectric power.
tions decisively influence the function S(T) by renormalizing the electronic spectrum. In fact, calculations based on Eqs. ~2! and ~12! in the static limit ( v !T) predict a change in the sign of the function S(T) at T'300 K and a maximum value for the thermoelectric power of FeSi equal to 230 mV/K at 253 K, which does not agree with Ref. 14. At the same time calculations based on Eqs. ~3!, ~12!, and ~13! ~within the framework of dynamic spin-fluctuation theory! give a maximum of S(T) near 50 K of roughly 750 mV/K ~see Fig. 1!. In this case, as we pass through the temperature range where the energy gap disappears and the number of mobile charge carriers increases abruptly, the paramagnon contribution becomes first comparable to and then considerably smaller than S d (T). In conclusion, it should be noted that small deviations from the stoichiometric composition of FeSi give rise to a sizable change in the magnitude of the thermoelectric power.14 In Ref. 15, Jarlborg attempted to analyze these features in terms of spin fluctuations in the static approximation. Although he discussed only Fe12x Si11x alloys with x!1, he found a considerable disagreement between the results of his calculations and values of the thermoelectric power observed in experiment, especially in the temperature range above
This is because at low temperatures the phonon heat capacity is proportional to T 3 , the paramagnon heat capacity to T.
1
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