Magnetic tricritical point and nematicity in FeSe under pressure

Magnetism induced by external pressure ($p$) was studied in a FeSe crystal sample by means of muon-spin rotation. The magnetic transition changes from second order to first order for pressures exceeding the critical value $p_{\mathrm{c}}\simeq 2.4–2.5$ GPa. The magnetic ordering temperature ($T_{\mathrm{N}}$) and the value of the magnetic moment per Fe site ($m_{\mathrm{Fe}}$) increase continuously with increasing pressure, reaching $T_{\mathrm{N}}\simeq 50$ K and $m_{\mathrm{Fe}}\simeq 0.25{\mu }_{\mathrm{B}}$ at $p\simeq 2.6$ GPa, respectively. No pronounced features at both $T_{\mathrm{N}}\left(p\right)$ and $m_{\mathrm{Fe}}\left(p\right)$ are detected at $p\simeq p_{\mathrm{c}}$, thus suggesting that the stripe-type magnetic order in FeSe remains unchanged above and below the critical pressure $p_{\mathrm{c}}$. A phenomenological model for the $(p,T)$ phase diagram of FeSe reveals that these observations are consistent with a scenario where the nematic transitions of FeSe at low and high pressures are driven by different mechanisms.

Figure 1

(a) The zero-field muon-time spectra collected at pressures $p\simeq 1.72$, 2.21, and 2.64 GPa. The red lines are fits of Eq. (1), with the sample contribution described by Eq. (2) to the experimental data. In order to increase the counting statistics, the ZF-$\mu \mathrm{SR}$ spectra accumulated in the temperature range from $\simeq 3$ to 20 K were summed together. (b) The Fourier transform of the ZF-$\mu \mathrm{SR}$ data shown in panel (a). The red lines are Lorentzian fits.

Figure 2

(a) Temperature dependence of the internal field $B_{\mathrm{int}}$ at $p=1.72$ and 2.21 GPa. (b) The same as in (a) but for $p=2.58$ and 2.64 GPa. The lines in (a) and (b) are fits of Eq. (4) to $B_{\mathrm{int}}\left(T\right)$ data. During the fit the data in (b) were combined together. (c) Temperature dependence of the magnetic volume fraction $m^{\mathrm{ZF}}$ at $p=1.72$ and 2.21 GPa. (d) The same as in (c) but for $p=2.58$ and 2.64 GPa. The temperature $T_{\mathrm{m}\sim 1}$ (the temperature where the magnetic volume fraction starts to deviate from unity) is determined as a crossing point of $m^{\mathrm{ZF}}=1$ line with the linear fit of $m^{\mathrm{ZF}}\left(T\right)$ in the vicinity of the magnetic transition [(d)].

Figure 3

(a) Pressure dependence of the magnetic ordering temperature $T_{\mathrm{N}}$ obtained from the fit of Eq. (4) to the experimental $B_{\mathrm{int}}\left(T\right)$ data. The $T_{\mathrm{N}}$ points for $p=2.58$ and 2.64 GPa correspond to temperatures where $B_{\mathrm{int}}\left(T\right)$ drops to zero [see Fig. 2]. (b) Pressure dependence of the zero-temperature value of the internal field. The blue stripe represents the pressure region where the magnetic transition changes from second order to first order, i.e., where a magnetic tricritical point exists.

Figure 4

(a) Temperature evolution of the magnetic volume fraction $m^{\mathrm{wTF}}$ of FeSe obtained in the wTF-$\mu \mathrm{SR}$ measurements at $p=2.05$ GPa. Closed and open symbols correspond to the experimental data obtained with increasing and decreasing temperature (the sweeping rate is $\simeq 0.2$ K/min, 5 min per data point). (b) The same as in (a) but for $p=2.58$ GPa.

Figure 5

Schematic $(p,T)$ phase diagram of FeSe used in the phenomenological calculation. The Pomeranchuk and magnetic (SDW) transitions meet at the multicritical point at $\left(p^{*},T_0\right)$. The respective transition temperatures are thus modeled as $T_p=T_0+\delta$ and $T_m=T_0-\delta$, with $\delta \propto p^{*}-p$ such that $\delta >0$ refers to the Pomeranchuk side and $\delta <0$ refers to the SDW side.

Figure 6

Value of the ordered magnetic moment on the Fe site $m_{\mathrm{Fe}}$ as a function of the magnetic ordering temperature $T_{\mathrm{N}}$. The red circles correspond to the results of this study on 101-oriented FeSe crystal. The open squares and the up and down triangles are from $B_{\mathrm{int}}\left(T_{\mathrm{N}}\right)\mu \mathrm{SR}$ measurements of Bendele et al. [21, 22] on polycrystalline FeSe samples synthesized by two different techniques. The gray stripe corresponds to the Mössbauer results of Kothapalli et al. [39] on high-quality single-crystalline FeSe sample. $T_{\mathrm{N}}$'s for Mössbauer data were obtained by fitting Eq. (4) to the temperature dependence of the hyperfine field $H_{\mathrm{pf}}$ at $p=2.5$ and 3.5 GPa [Fig. 2(b) in Ref. 39]. The solid lines are guides for the eyes.

Figure 7

Dependence of the magnetic ordering temperature $T_{\mathrm{N}}$ and the temperature where the magnetic volume fraction starts to decrease below unity $T_{\mathrm{m}\sim 1}$ on applied pressure $p$ in FeSe obtained by different techniques. The red closed and semiopen circles correspond to the results of this study on 101 oriented FeSe crystal. The open stars are $\mu \mathrm{SR}$ data on FeSe polycrystalline samples [21, 22]. Open squares and triangles are NMR and Mössbauer data from Refs. [39, 40, 41], respectively. The open circles correspond to $T_{\mathrm{N}}$ obtained in resistivity experiments [58, 59, 60]. Lines are guides for the eyes.

Figure 8

Value of the ordered magnetic moment on the Fe site $m_{\mathrm{Fe}}$ as a function of the magnetic ordering temperature $T_{\mathrm{N}}$ and on the temperature where the magnetic volume fraction starts to decrease below unity $T_{m\sim 1}$. The lines are guides for the eyes. $m_{\mathrm{Fe}}$ tends to zero value at $T_{\mathrm{N}}(m_{\mathrm{Fe}}=0)\sim 18$ K, while no temperature offset is expected for $T_{m\sim 1}(m_{\mathrm{Fe}}=0)$.