Magnetic Measurements of Narrow-Gap Semiconductor FeSb₂ under High Pressure

Abstract

Pressure effects on magnetic susceptibility χ and energy gap E_g⁰ of narrow-gap semiconductor FeSb₂ were investigated for the temperature range of 50–300 K and pressures up to 13 kbar. The estimated E_g⁰ for ambient pressure, E_g0(0), was 29 meV. By application of pressure, χ was suppressed, and E_g0(P) was estimated to be 52 meV for 13 kbar. The fourth-order expansion coefficient γ of the free energy in magnetization was positive and enhanced by applying pressures.

Fig. 1 Temperature dependence of the magnetic susceptibility of FeSb₂ at various pressures.

1. Introduction

Fe-based narrow-gap semiconductors such as FeSb₂ and FeSi have attracted much interest because of the intriguing behavior of its transport, thermodynamic and magnetic properties.^1–5) In particular, FeSb₂ has a large Seebeck coefficient in the vicinity of 10 K, so that FeSb₂ is considered to be one of the candidates of cryorefrigeration at low temperature.⁵⁾ FeSb₂ and FeSi also have been studied from a view point of Kondo insulator system.^4,6,7)

Magnetic susceptibility χ of FeSi with a cubic structure shows an activation-type increase with increasing temperature T above 70 K.^2,3) After showing a broad maximum on χ around 500 K, the χ-T curve exhibits the Curie-Weiss like decrease.^2,3) In order to explain the χ-T curve of FeSi, a negative mode-mode coupling mechanism was proposed based on the spin fluctuation theory,⁸⁾ which affects the magnetization process. This theory suggested that the fourth-order expansion coefficient γ of the magnetic free energy in magnetization was negative value. An early χ-T measurement of FeSi was in qualitative agreement with the theoretical assumption.³⁾

Precise magnetization measurements on FeSi showed that γ has a positive value at low temperature and decreases by the order of 10⁵ with increasing T.⁹⁾ Takahashi gave a new theoretical explanation to the M-H curve of FeSi based on the spin fluctuation theory without the negative mode-mode coupling mechanism.^10,11) For magnetic properties of FeSi, he employed a simple density of states curve with an energy gap E_g and considered quantum spin fluctuations together with thermal spin fluctuations. Using his model, E_g (∼ 62 meV) of FeSi and its pressure dependence were evaluated from the χ-T curves for various pressures P, which showed that E_g increased with increasing P.¹²⁾ This result is qualitatively consistent with results estimated from electrical resistivity ρ data.¹²⁾

χ of FeSb₂ also shows an activation-type increase with increasing T above 50 K. After showing a broad maximum on χ around 300 K, the χ-T curve exhibits the Curie-Weiss like decrease, which is similar to the behavior of FeSi. Koyama et al. analyzed the M-H curves of a FeSb₂ single crystal based on Takahashi’s spin fluctuation theory and estimated γ of FeSb₂.¹³⁾ FeSb₂ also showed that the positive γ decreases by the order of 10⁶ with increasing T.¹³⁾

From a viewpoint of Takahashi’s theory, E_g and the spin fluctuation play an important role for the magnetic properties of FeSb₂ as well as FeSi. When a pressure is applied to FeSb₂, E_g will change and the spin fluctuation and magnetization process will be modified. The pressure dependence of E_g of FeSb₂ was already estimated by ρ measurements.¹⁴⁾ However, to the best of our knowledge, there is no report on the pressure effect of E_g and the detailed magnetic properties of FeSb₂. It is difficult to measure precisely the magnetization under pressures, because the magnetic moment (χ ∼ 2 × 10⁻⁶ emu/g at 300 K)¹³⁾ of FeSb₂ is smaller than that of FeSi.

In this study, we have measured χ-T and M-H curves of a FeSb₂ single crystal for P ≤ 13 kbar in order to examine the pressure effects on the energy gap and γ evaluated from magnetic properties based on Takahashi’s spin fluctuation theory.

2. Experimental Procedure

Single crystals of FeSb₂ were grown by an Sb self-flux method using pure elements (Fe, 4N; Sb, 5N). The sample preparation was presented in a previous paper.¹³⁾ The crystal was cut into a pillar with the size of 1.6 × 1.6 × 2.2 mm³ along the a-axis. The magnetizations M were measured along the a-axis for magnetic fields H up to 50 kOe and P ≤ 13 kbar for 50 ≤ T ≤ 300 K using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design) and a clamp-type pressure cell. The pressure cell made of a Cu–Be alloy was used. The M data were corrected by subtracting the background magnetization of the pressure cell. The original design of the cell and the experimental technique were presented in Ref. 15) in detail.

3. Results and Discussion

Figure 1 shows the χ-T curves at various pressures. Here, χ for P = 0 kbar and for high pressures was deduced from the M-T measurements under H = 1 kOe. For P = 0 kbar, χ is almost zero at 50 K and shows an activation-type increase with increasing T. At 300 K, χ is 5.5 × 10⁻⁴ emu/mol, which is consistent with the previous report.¹⁰⁾ By application of P, χ is suppressed.

Fig. 1

Temperature dependence of the magnetic susceptibility of FeSb₂ at various pressures.

The results are discussed using a simple semiconductor-model based on the Takahashi’s theory.⁹⁾ The itinerant electrons in the conduction band behave like non-interacted moments due to the effect of spin fluctuations, and χ of each electron is proportional to 1/T in the temperature region with the activation-type behavior. We assume that χ of FeSb₂ comes mainly from thermally excited electrons and holes. The numbers of the excited electrons and holes, n_e and n_h, are equal to each other and are proportional to T^3/2 exp(−E_g/2k_BT), where E_g and k_B are the energy gap and the Boltzmann constant, respectively. The susceptibility of each electron (hole), χ_e (χ_h), is proportional to 1/T. Assuming that the temperature dependence of E_g is approximately given by E_g = E_g⁰ + αT, the total susceptibility χ = n_eχ_e + n_hχ_h = n_e (a/T + b/T) is rewritten as   

\begin{equation} \chi = AT^{1/2} \exp (-E_{\text{g}}{}^{0}/2k_{\text{B}}T), \end{equation} (1)

where E_g⁰ is the energy gap for 0 K, and α, a, b and A are parameters. Figure 2 shows the plots of ln(χT^−1/2) vs. 1/T at various pressures for the 170 ≤ T ≤ 220 K. The plots yield straight lines, indicating that the χ data satisfy eq. (1). The values of E_g⁰ were estimated from the slope of these lines.

Fig. 2

Plots of ln(χT^−1/2) vs. 1/T at various pressures.

Figure 3 shows the pressure dependence of E_g⁰. In this figure, the reported data¹¹⁾ for FeSb₂ are also plotted. Our value of E_g⁰ for P = 0 kbar, E_g⁰(0), is 29 meV, which is consistent with the results of the reported ρ measurements.^14,16) This suggests that the above-mentioned model can be applicable to FeSb₂. This E_g⁰(0) is smaller than that (62 meV) determined by same Takahashi’s model for FeSi. By application of pressure, E_g⁰(P) was enhanced and was estimated to be 52 meV for P = 13 kbar. The obtained values under pressures are larger than those deduced from the ρ measurement.¹⁴⁾ This is probably due to an influence of the background or pressure-transmitting medium of the clamp-type pressure cell. However, it seems that both our data (5 ≤ P ≤ 13 kbar) using the pressure cell and the results of ρ measurements increase linearly with increasing P. This indicates that the contribution of the background is almost constant.

Fig. 3

Pressure dependence of the energy gap estimated at T = 0 K. The data deduced from the electrical resistivity measurements by Takahashi et al. (open circles)¹⁴⁾ and Bentien et al. (open squares)¹⁶⁾ are also plotted for comparison. The broken line is calculated by the least-square method using a linear function for data under pressures.

The relative values of [E_g⁰(P) − E_g⁰(0)]/E_g⁰(0) are plotted as a function of pressure in Fig. 4. Here, E_g⁰(0) (= 43.2 meV) was estimated from the linear extrapolation calculated by the least-squares method for our data under 5 ≤ P ≤ 13 kbar. The values estimated from the ρ measurements for 1 bar ≤ P ≤ 18 kbar¹⁴⁾ are in agreement with our data which are expressed as d ln E_g⁰/dP = 17 × 10⁻⁶ bar⁻¹. This value is almost twice as large as that (9.6 × 10⁻⁶ bar⁻¹) of FeSi,¹²⁾ as shown in Fig. 4.

Fig. 4

Pressure dependence of [E_g⁰(P) − E_g⁰(0)]/E_g⁰(0). The data deduced from the electrical resistivity measurements of FeSb₂ by Takahashi et al. (open squares)¹⁴⁾ and from the magnetic susceptibility measurements of FeSi by Koyama et al. (open circles)¹²⁾ are also plotted for comparison. The solid and broken lines are calculated by the least-square method using a linear function for data of this work and of FeSi, respectively.

Figure 5 shows the M-H curves in fields up to 50 kOe under pressures of a zero (ambient pressure) (a) and 10 kbar (b) above 150 K. The magnitude of M for the pressures of 0 and 10 kbar were corrected using the reported value¹³⁾ for the a-axis of an FeSb₂ single crystal at each temperature to remove the paramagnetic background of the pressure cell. The data clearly showed that the paramagnetic M-H curves were suppressed by application of pressure.

Fig. 5

Magnetization curves in fields up to 50 kOe under pressures of 0 kbar (ambient pressure) (a) and 10 kbar (b) above 150 K.

In order to examine the presence of the magnitude of the non-linear term of the M-H curves, the following Arrott plot analysis is very suited.^12,13) The magnitude of the induced moment of FeSb₂ is small even in the external magnetic field in all the present experimental situations.^12,13) The nonlinear M-H curves is, therefore, well described by the following Landau expansion form of the magnetic free energy in powers of the M up to the forth-order term,   

\begin{equation} F(M) = \frac{1}{2\chi}{M^{2}} + \frac{1}{4}\gamma M^{4}, \end{equation} (2)

where χ is the magnetic susceptibility and the coefficient γ represents the 4-th order expansion coefficient. From the thermodynamic relation, the following magnetic equation of state is derived:   

\begin{equation} \frac{\partial F(M)}{\partial M} = \frac{M}{\chi} + \gamma M^{3} = H. \end{equation} (3)

Or it is represented in terms of the following relation:   

\begin{equation} M^{2} = \frac{1}{\gamma}\frac{H}{M} - \frac{1}{\gamma \chi}. \end{equation} (4)

This means that if we plot the observed values of M² against those of H/M (i.e. in the form of the Arrott plot), we can expect the good linearity of the plot. The value of γ was estimated from the slope of the M²-H/M plot. Figure 6 shows the Arrott plot analysis of the M-H curves at various temperatures above 150 K for pressures of 0 kbar (a) and 10 kbar (b). A good linearity was satisfied up to highest magnetic field for the data of both pressures.

Fig. 6

Arrott plots of FeSb₂ at various temperatures above 150 K for pressures of 0 kbar (a) and 10 kbar (b).

Figure 7 shows the temperature dependence of γ for pressures of 0 kbar and 10 kbar. In this figure, the reported data¹³⁾ for the a-axis of a FeSb₂ single crystal under ambient pressure was also presented. As seen in this figure, the obtained data of 0 kbar was almost consistent with the previous data in ambient pressure. That is, the value of γ is very large at low temperature, and it decreases significantly with increasing temperature. By applying a pressure of 10 kbar, γ was enhanced and about ten times larger than that of 0 kbar. According to the results of FeSi reported by Takahashi^10,11) and Koyama et al.,⁹⁾ if we evaluate the M-H curve based on the single particle picture with the Fermi level in the middle of the narrow semiconducting energy gap as shown by band structure calculation, we will generally predict the negative γ value, for it is related with the curvature of the density of states curve around the Fermi energy. However, our results show that E_g⁰ of FeSb₂ for 50–300 K temperature range was enhanced by application of a pressure, and the value of γ was positive and enhanced. This suggests that the magnetic behavior of FeSb₂ is not only described by a simple single-excitation picture with the narrow energy gap but also considered based on spin fluctuation.

Fig. 7

Temperature dependence of the fourth order coefficient γ of the free energy for pressures of 0 kbar and 10 kbar.

4. Conclusions

We studied the pressure effects on the magnetic susceptibility χ and the energy gap E_g⁰ of FeSb₂ for the 50–300 K temperature range. By application of the pressure, χ is suppressed and E_g⁰ was enhanced with d ln E_g⁰/dP = 17 × 10⁻⁶ bar⁻¹. The energy gap of FeSb₂ is sensitive to the application of high pressure compared with that of FeSi. The fourth-order expansion coefficient γ of the free energy was positive and enhanced by applying pressures.

Acknowledgment

This work was supported in part by the KAKENHI 22360285. The magnetic susceptibility measurements under high pressures were carried out in Institute for Solid State Physics, the University of Tokyo.

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