Magnetotransport studies of the Sb square-net compound LaAgSb₂ under high pressure and rotating magnetic fields

First, we show the experimental results obtained under high pressure and rotating magnetic fields. In this study, the magnetic field was tilted from the [001] axis to the normal directions to discuss the dimensionality of the Fermi surface. $\theta$ is defined as a field angle measured from the [001] axis. We utilized the term “CDW1+2 phase” ($0<P<P_{CDW2}$, CDW1 and CDW2 coexists), “CDW1 phase” ($P_{CDW2}<P<P_{CDW1}$, only CDW1 survives), and “normal metallic phase” ($P>P_{CDW1}$, no CDW exists) to specify the electronic state. As mentioned in the previous report Akiba et al. (2021), the amplitude of the SdH oscillation in the CDW1 phase does not obey the conventional Lifshitz-Kosevich formula and takes the maximum at a relatively higher temperature. Thus, most data were taken at 20 K in the CDW1 phase, unless otherwise specified.

In Fig. 1, we show the field-angular dependence of the in-plane magnetoresistance ($\rho_{100}$ and $\rho_{110}$), where electrical current ($I$) flows along the [100] and [110] directions. In these measurements, the magnetic field ($B$) was rotated in the plane perpendicular to $I$. At all phases, remarkable changes in the magnetoresistance were observed depending on the field direction. We also observed an angular dependence of the SdH oscillation, which will be discussed later. In the CDW1+2 and CDW1 phases, the angular dependence of the MR is rather simple: it shows the largest positive MR effect in the case of $B\parallel[001]$ (light orange traces in Fig. 1), and then gradually decreases as $B$ approaches the in-plane [100] or [110] directions (dark blue traces in Fig. 1). In the normal metallic phase, however, the MR effect takes the maxima at intermediate angles as $\theta$ increases, and then, takes the minimum around $\theta=90^{\circ}$.

This change in the MR is clearly visible in the polar plots shown in Fig. 2(a) and (b). In Fig. 2(a) and (b), the radius data represent the magnetoresistance at 8 T normalized by the value at 0 T defined by $\Delta\rho_{100,110}/\rho\equiv\rho_{100,110}(8T)/\rho_{100,110}(0T)-1$. In the CDW1+2 and CDW1 phases, it shows simple and almost identical two-fold symmetry in both current configurations, and the polar spectra are insensitive to the current direction.

In the normal metallic phase, in contrast, polar spectra show a more complicated structure and obviously depend on the current direction. The polar patterns show butterfly-like shapes in both cases, which are similar to those observed in the other square-net systems Ali et al. (2016); Lv et al. (2016); Hu et al. (2016); Kumar et al. (2017); Voerman et al. (2019); Novak et al. (2019); Chiu et al. (2019); Shirer et al. (2019). In case of $I\parallel$ [100], the magnetoresistance takes broad local maxima around $\pm 45^{\circ}$. In case of $I\parallel$ [110], we can recognize the distinctive local maxima around $\pm 30^{\circ}$ and $\pm 60^{\circ}$. In contrast to CDW1+2 and CDW1 phases, $\theta=0^{\circ}$ changes into the local minima in both configurations.

In Fig. 2(c), we summarize the $\theta$ dependence of $n$, assuming the field dependence to be $\Delta\rho_{100,110}/\rho\propto B^{n}$. Around the local maxima of $\Delta\rho_{100,110}/\rho$, $n$ is slightly enhanced in both cases. At $\theta=90^{\circ}$, in contrast, $n$ considerably decreases, and even shows almost linear field dependence. At any $\theta$, $n$ is significantly smaller than conventional $n=2$.

The drastic change in the magneto-transport properties at $P_{CDW1}$ mentioned above is evidently accompanied by emergence of a Fermi surface, which has been gapped out by CDW1. Because the angular dependence of the magnetoresistance roughly reflects the symmetry of the Fermi surface cut perpendicular to the current direction Zhang et al. (2019), the two-fold symmetry is expected to be trivial considering the tetragonal crystal structure. Additional structures observed only in the normal metallic phase are expected to contain information of the Fermi surface geometry. This behavior will again be discussed in later sections.

Next, we focus on the angular dependence of the out-of-plane magnetoresistance ($\rho_{001}$). The in-plane crystal orientation has not been identified for this sample. $\theta=0^{\circ}$ and $\theta=90^{\circ}$ correspond the longitudinal ($I\parallel B$) and transverse ($I\perp B$) configurations, respectively.

Figure 3(a), (b), and (c) show $\rho_{001}$ under various field directions in CDW1+2, CDW1, and normal metallic phases, respectively. The angular dependence of $\Delta\rho_{001}/\rho$ at 8 T is summarized in Fig. 3(d). Common to all phases, the magnetoresistance is considerably small and even decreases as $B$ increases when $B\parallel[001]$ (light orange), whereas it is robustly enhanced around $\theta=90^{\circ}$ (dark blue). This behavior introduces very sharp peaks at around $\theta=90^{\circ}$ in Fig. 3(d). $\Delta\rho_{001}/\rho$ at $\theta=90^{\circ}$ is smaller in the CDW1+2 phase and becomes approximately two times larger in the CDW1 and normal metallic phases. The value of $\Delta\rho_{001}/\rho$ is not substantially different between the CDW1 and normal metallic phases. Figure 3(e) and (f) show $\rho_{001}$ at representative pressures around $|\theta|\sim 0^{\circ}$ and $90^{\circ}$, respectively. For $|\theta|\sim 0^{\circ}$, the characteristic negative magnetoresistance is well consistent with our previous result Akiba et al. (2021). For $|\theta|\sim 90^{\circ}$, in contrast, negative contribution is completely absent, and $\rho_{001}$ increases with the application of $B$ without saturation.

Then, we move onto the angular dependence of the SdH oscillations superimposed on $\rho_{100}$, $\rho_{110}$, and $\rho_{001}$. Figure 4 shows the angular dependence of the SdH frequency ($F$). The current and field configurations are illustrated at the bottom of each column. The corresponding raw data are shown in Fig. S3 in Supplemental Material SM_ .

At ambient pressure (the first row in Fig. 4), we observed two frequencies, $F\sim 160$ T and $\sim 440$ T, at $\theta=0^{\circ}$, which correspond to the $\beta$ and $\gamma$ branches, respectively, in the previous studies Myers et al. (1999a); Inada et al. (2002); Bud’ko et al. (2008). Each frequencies increases as the magnetic field is tilted from [001] axis, which is consistent with the previous quantum oscillation measurements at ambient pressure Myers et al. (1999a); Inada et al. (2002).

In the CDW1 phase (the second row in Fig. 4), we detected a single frequency of 10 T at $\theta=0^{\circ}$, which is consistent with our previous report Akiba et al. (2021). The angular dependence of this frequency well obeys the $1/\cos\theta$-scaling over the wide $\theta$ range (up to $\theta=60^{\circ}$), indicating the cylindrical geometry. This indicates the existence of a considerably narrow cylindrical Fermi surface in the CDW1 phase. We also confirmed that the $\theta$ dependence of $F$ is almost identical at the lower temperature of 1.6 K, as shown by the open markers in Fig. 4(h).

In the normal metallic phase (the third row in Fig. 4), the SdH oscillation has a single component in all configurations, whose frequency $F\sim 50$ T at $\theta=0^{\circ}$ is consistent with that of the $\omega$ branch identified in the previous study Akiba et al. (2021). The obtained data are relatively poor compared to those in the CDW1 phase because of the significant decay of the oscillation amplitude as $\theta$ increases. In the case in which the cross section comes from an elongated ellipsoid characterized by shorter ($a$) and longer ($b$) axes, the $\theta$ dependence is proportional to $1/\sqrt{\cos^{2}\theta+e^{-2}\sin^{2}\theta}$ where $e=b/a$. The above function shows similar $\theta$ dependences at a small $\theta$. We show the expected $\theta$ dependence when $e=2.9$ in Fig. 4(c), (f), and (i) by red broken lines. For the $\omega$ branch, it is difficult to discuss the geometry only from these data. Thus, we focus on the low-temperature MR measured at 0.24 K for $\theta=90^{\circ}$, which is shown in Fig. 5. When the SdH oscillation comes from a cylindrical geometry, the orbit is open when $\theta=90^{\circ}$, and thus, no SdH oscillation is expected. In case of ellipsoidal geometry, in contrast, there exists an extremum cross section at $\theta=90^{\circ}$, and thus, it is reasonable to expect an SdH oscillation under sufficiently small thermal damping. As seen in Fig. 5, oscillation is absent at the lowest temperature of the present study, and hence, we regard the $\omega$ branch resulting from a cylindrical geometry.

To summarize the SdH measurements, we conclude that Fermi surfaces detected in the high pressure phases seem to be cylindrical along the $z$ direction. This fact indicates that the strong two-dimensional nature is maintained up to the normal metallic phase.

Here, we discuss the Fermi surface in the normal metallic phase based on the band structure calculations. Because accurate information on the crystal structure and atomic coordinates etc within the CDWs is not available, it is difficult to quantitatively discuss the electronic structure in the CDW1+2 and CDW1 phases. In this study, therefore, we mainly focus on the normal metallic phase, in which we can directly compare the calculation with experiments owing to the absence of the energy gap and band reconstruction by CDW.

Figure 6(a) shows the band structure at 3.5 GPa (black) and ambient pressure (red). We can see that at a pressure of 3.5 GPa, no remarkable change occurs near the Fermi level, such as the Lifshitz transition. Figure 6(b-e) shows the Fermi surface of LaAgSb₂ at 3.5 GPa with the absolute value of the Fermi velocity $v_{F}$ shown by color code. Hereafter, we refer to the bands constructing the surfaces shown in Fig. 6 (b), (c), (d), and (e) as bands 1, 2, 3, and 4, respectively. Bands 1 and 2 (3 and 4) are hole (electron) surfaces enclosing the higher- (lower-) energy region within the surface. The hot spot of $v_{F}$ (e.g., around $X$ and on the way of $\Gamma$-$M$ and $Z$-$A$) corresponds to the point at which almost linearly dispersed bands cross $\epsilon_{F}$ in Fig. 6(a). The maximum $v_{F}$ of $\sim 1.6\times 10^{6}$ m/s is consistent with the previous calculation Ruszala et al. (2020) and comparable to those of SrMnBi₂ and graphene Park et al. (2011). Compared with the previous calculation by Myers et al. Myers et al. (1999a) which is often referred in the discussion of quantum oscillation and nesting properties in this material, the shape of bands 2-4 is almost identical. However, our calculation supports that band 1 is open along the $z$ direction, which is a dice-like closed pocket in the calculation by Myers et al. The present Fermi surface geometry is well in accordance with a recent computational study reported by Ruszala et al. Ruszala et al. (2020). In Tab. 1, we list the calculated cross section $S$ and the corresponding SdH frequency $F$ of several orbits at 3.5 GPa and ambient pressure.

We have shown in the experimental results that there exists a cylindrical Fermi surface with a small cross-section of $\sim$50 T in the normal metallic phase. In our calculation, the orbit C1 around the $Z$ point in band 1 (Fig. 6(g)) can cause $1/\cos\theta$-type angular dependence, whose frequency is calculated to be 49.9 T at 3.5 GPa (Tab. 1). This value shows reasonable agreement with the observed frequency. Although orbit C2 in Fig. 6(f) can also cause $1/\cos\theta$-type dependence, the frequency is calculated to be 486 T, which is considerably larger than the observed frequency. The orbit C3 in band 4 (Fig. 6(f)) shows comparable cross section with C1. However, the complete absence of the SdH shown in Fig. 5 might be unlikely for the ellipsoidal shape. The other cross-sections are considerably larger, and thus, we can exclude them from consideration.

According to the above results, we conclude that experimentally observed cross-section in the normal metallic phase originates from orbit C1, i.e., our result supports the cylindrical Fermi surface for band 1. Conventionally, the CDW2 has been considered to be caused by the $k_{z}$-oriented nesting vector within band 1 Song et al. (2003), which was deduced from the dice-like shape of the Fermi surface. Our conclusion suggests a careful reconsideration whether this nesting picture is valid in the case of the present Fermi surface geometry.

Here, we show the computational results of the conductivity and resistivity tensors in the normal metallic phase in $B\parallel[001]$ and compare them with the experimental results. In the following calculation, the effect of Landau quantization is ignored. This assumption is assumed to be reasonable because the present situation is far from the quantum limit state.

Figures 7 (a-c) show the band-resolved conductivity components $\sigma_{ij}^{(n)}/\tau_{n}$ at 3.5 GPa. Here, the weight-averaged velocity in Eq. (3) is arranged as a function of $B\tau_{n}$ as

and hence, $\sigma_{ij}^{(n)}/\tau_{n}$ is represented as a function of $B\tau_{n}$ as

The relaxation time in the $n$-th band ($\tau_{n}$) appears as an unknown parameter in the form of $\sigma_{ij}/\tau_{n}$ or $B\tau_{n}$ in the results shown in Fig. 7(a-c). We recognize that in all components, the dominant contribution comes from band 3. Based on Eq. (6), the dominance of band 3 is explained by the largest surface area and the distribution of high $v(\bm{k})$ [Fig. 6(d)] owing to its steep linear dispersion. As shown in Fig. 7(b), bands 1 and 2 (band 4) lead to positive (negative) contributions on the Hall conductivity, because these Fermi surfaces are hole (electron) surfaces and host only closed orbits when sliced by the (001) plane. In contrast, Band 3 has a remarkable positive contribution, although it is referred to in the conventional terminology as an electron surface. The reason for this behavior is assumed to be the characteristic hollow-like geometry that enables the formation of hole-type closed orbits when sliced by the (001) plane (e.g., C4 and C5 in Fig. 6(f)). The calculated $\sigma_{xy}/\tau$ indicates that LaAgSb₂ in the normal metallic phase with $B\parallel[001]$ is far from the compensation condition, and the orbital character considerably inclines toward the hole-type.

By summing all band-resolved components, we obtain the total conductivity tensor divided by the relaxation time ($\sigma_{ij}/\tau=\sum_{n=1}^{4}\sigma_{ij}^{(n)}/\tau_{n}$), whose non-zero components are shown in Fig. 7(d-f). Here, we assume that the relaxation time is independent of the band index, namely, $\tau_{n}=\tau$ for all four bands. We can see in the insets of Fig. 7(d) and (e) that $\sigma_{xx}\propto B^{-2}$ and $\sigma_{xy}\propto B^{-1}$ are satisfied, which are generally known as the field dependence in the uncompensated case with no open orbit Chambers (1990). By calculating the inverse matrix of $\sigma_{ij}/\tau$, we finally obtain the resistivity tensor $\rho_{ij}\tau$, which can be compared with the experimental data.

In uncompensated metal with no open orbit, the Hall resistivity no longer depends on the details of relaxation time and can be determined only by the degree of compensation between electron-like and hole-like orbits in the high-field limit. Thus, the Hall resistivity is regarded to be a quantitative index of the Fermi surface geometry. Here, we compare and discuss the experimental and theoretical Hall responses in the normal metallic phase.

Fig. 7(h) shows the calculated Hall response $\rho_{yx}\tau$ as a function of $B\tau$. The black curve is total $\rho_{yx}\tau$, calculated by considering contributions from all bands. It shows almost linear magnetic-field dependence up to $B\tau=40$ T ps. This linear dependence is consistent with the experimental results Akiba et al. (2021), and thus, we can quantitatively compare the Hall coefficient $R_{H}$. From the slope of the $\rho_{yx}\tau$-$B\tau$ plane, we can theoretically evaluate $R_{H}=\rho_{yx}\tau/(B\tau)$, which is a $\tau$-independent quantity. We obtained $R_{H}=0.903\times 10^{-9}$ m³/C from Fig. 7(h), which shows reasonable agreement with the experimental value $R_{H}=1.16\times 10^{-9}$ m³/C at 3.4 GPa Akiba et al. (2021). This agreement supports the validity of our Fermi surface model shown in Fig. 6.

We can also calculate the “partial” Hall resistivity, in which the contribution from a specific band is intentionally excluded when calculating $\bm{\rho}$. This virtually describes the case in which a specific band is absent due to full CDW gap. In Fig. 7(h), we show the partial Hall resistivity without the $n$-th band (w/o 1 to 4). Obviously, the absence of bands 1, 2, and 4 hardly affects the slope of $\rho_{yx}\tau$, and only hollow-like band 3 causes significant changes on $\rho_{yx}\tau$. Previous experiments have shown that $\rho_{yx}$ at 9 T is reduced to 1/8 across $P_{CDW1}$ Akiba et al. (2021). Such a large change can only be explained by the disappearance of band 3. Thus, the above discussion confirms the dominant role of band 3 for the formation of CDW1. However, we note that experimental $\rho_{yx}$ in the CDW1 phase shows unignorable non-linearity in the weak-field region, which is not reproduced by the calculated partial Hall resistivity. In-depth knowledge regarding the Fermi surface within the CDW phase is indispensable to enter further quantitative aspects of the Hall response below $P_{CDW1}$.

Next, we move onto the transverse ($\rho_{xx}$) and longitudinal ($\rho_{zz}$) magnetoresistance. Figure 7(g) and (i) shows $\rho_{xx}\tau$ and $\rho_{zz}\tau$ as a function of $B\tau$, respectively. $\rho_{xx}\tau$ increases at the weak-field region and then shows almost linear $B\tau$ dependence with a considerably small slope in $B\tau>10$ T ps. $\rho_{zz}\tau$ also shows a sudden increase at a low $B\tau$ region and becomes almost flat above $B\tau>5$ T ps. In the experimental result, in contrast, both $\rho_{xx}$ and $\rho_{zz}$ show considerable field dependence: $\rho_{xx}$ is proportional to $B^{\sim 1.5}$ as shown in Fig. 2(c), and $\rho_{zz}$ shows characteristic negative longitudinal magnetoresistance as shown in Fig. 3(e).

Here, we discuss the reason for the mismatch between the experimental and calculated results. The possible factor is the relaxation-time approximation adopted in this calculation. We ignored the details of $\tau$, namely, (1) $n$ dependence of $\tau$ and (2) $\bm{k}$ dependence of $\tau$. As for (1), we can recognize, in the case of $\sigma_{zz}$, that irrespective of how we set $\tau_{n}$ individually, we cannot reproduce the decrease in $\rho_{zz}\sim 1/\sigma_{zz}$ observed in the experiment. Thus, the factor (1) alone cannot explain the mismatch between the experiment and calculation. To consider factor (2), we have to introduce the $\bm{k}$-dependent relaxation time $\tau_{n}(\bm{k})$ in the integrant in Eq. (1). Recent theoretical work on elemental metals showed that there exists remarkable $\bm{k}$ dependence of $\tau$ even in such simple metals when the electron-phonon interaction is considered Mustafa et al. (2016); Rittweger et al. (2017). Because we do not know the precise representation of $\tau_{n}(\bm{k})$, we cannot discuss the effect of (2) quantitatively. In the scope of the present study, factor (2) is not excluded from the possible cause. Unlike the case of the Hall response, the contribution of the relaxation time to the transverse and longitudinal magneto-transport does not vanish in the high-field limit, and thus, they are affected by the details of the scattering process.

The above consideration is also experimentally supported by the remarkable violation of the so called Kohler’s rule in the normal metallic phase. When the relaxation time is independent of $n$ and $\bm{k}$, the transverse magnetoresistance should follow Kohler’s scaling rule. When we define $\Delta\rho/\rho\equiv[\rho(B,T)-\rho(0,T)]/\rho(0,T)$ where $\rho(B,T)$ represents the transverse magnetoresistance at magnetic field $B$ and temperature $T$, Kohler’s rule requires $\Delta\rho/\rho=f(B\tau)=F(B/\rho(0,T))$, where $f$ and $F$ represent arbitral functions, and we assume $\tau\propto 1/\rho(0,T)$. To test the validity of Kohler’s rule in the present case, we construct a so called Kohler’s plot for each phase, as shown in Fig. 8. Here, the magnetic field is applied along [001] direction, and current along an in-plane direction. In CDW1+2 and CDW1 phases, $\Delta\rho/\rho$ roughly falls into a universal curve as a function of $B/\rho(0,T)$ within the given temperatures, indicating reasonable agreement with Kohler’s rule. In the normal metallic phase, however, the deviation from the Kohler’s scaling becomes significant, indicating that the picture of isotropic $\tau$ should be refined in the normal metallic phase.

Although we mentioned above that the conductivity calculation based on the conventional relaxation-time approximation is inadequate for a quantitative discussion on $\rho_{xx}$ and $\rho_{zz}$, it is still insightful from the qualitative viewpoint to see whether the observed magneto-transport properties can arise from the Fermi surface geometry. Thus, we proceed with a comparison of the angular dependence of $\rho_{xx}$ and $\rho_{zz}$ between the experiment and calculation. We performed similar calculations shown in Fig. 7(g) and (i) for various $\theta$ with an interval of $2^{\circ}$ and constructed a polar plot.

Figures 9(a) and (b) show the calculated polar plot of $\Delta\rho_{100}/\rho$ and $\Delta\rho_{110}/\rho$, respectively. For several representative angles, the $B\tau$ dependence of $\Delta\rho_{100}/\rho$ and $\Delta\rho_{110}/\rho$ is also shown in Fig. 9(c) and (d). We can see that remarkable field-angular dependence of the MR can arise from the Fermi surface geometry. Further, we can recognize essential features observed in the experiment such as the local minima at $\theta=0^{\circ}$ and 90^∘ and a current-direction-dependent polar pattern. A notable feature in the calculated polar plot is the very sharp decrease of the MR at specific field angles, which results in a complex pattern. As mentioned in Fig. S4 in Supplemental Material SM_ , these dips seem to arise in a certain manner, which apparently resembles the Kajita–Yamaji oscillation observed in quasi-two-dimensional materials Kajita et al. (1989); Yamaji (1989). In the present stage, however, the specific mechanism of the reduction in resistance cannot be identified. The changes in the MR are more rapid in the case of $I\parallel[100]$, which occurs within $10^{\circ}$. Focusing on the case of $I\parallel[110]$, the calculated polar pattern shows two distinct “petals” indicated by arrows in Fig. 9(b). This feature qualitatively agrees with the experimental butterfly-like pattern shown in Fig. 2(b).

However, we should note several undeniable mismatches between the experimental and calculated results. First, the theoretical polar pattern for $I\parallel[100]$ seems to have little resemblance with the experimental one [Fig. 2(a)]. As shown in Fig. S5 in Supplemental Material SM_ , the Fermi surface shift within $\pm 50$ meV does not cause notable changes in the polar pattern. We assume that coarse experimental data ($\sim 10^{\circ}$ interval) and a slightly lower residual resistivity ratio (RRR) of the measured sample (RRR = 106 at 3.3 GPa for $I\parallel[100]$, whereas RRR = 152 at 3.3 GPa for $I\parallel[110]$) might mask the rapid change in the MR. Second, the details of the polar pattern, e.g., the position of the maxima/minima for the $I\parallel[110]$ case and the elongated shape of the calculated patterns disagree with the experimental results. Although a slight difference in the Fermi surface curvature or shape can contribute to these mismatches, we cannot specify the exact reason in the present study. Because there can be seen several proposals for the possible Fermi surface Arakane et al. (2007); Bud’ko et al. (2008); Hase and Yanagisawa (2014); Chen et al. (2017); Ruszala et al. (2020), which are slightly different from each other, the details of the geometry are open to further argument. In addition, anisotropic $\tau$ revealed in the previous section should also contribute to the above mismatches, and hence, more accurate treatment for $\tau$ is indispensable for a quantitative discussion.

Finally, we show in Fig. 9(e) the calculated $\Delta\rho_{001}/\rho$ as a function of $\theta$. In this calculation, $B$ directs [100] at $\theta=90^{\circ}$. We can see a very sharp peak at around $\theta=90^{\circ}$, and no remarkable structure is discernible. This behavior qualitatively reproduces the experimental results. The enhancement of MR around $\theta=90^{\circ}$ can be understood as the very small $v_{z}$ component because most cross-sections are open along the [001] direction when the Fermi surface is sliced parallel to the [001]. From Eq. (1), small $v_{z}$ results in small $\sigma_{zz}$, and thus, $\rho_{zz}$ becomes large.

To summarize, although complete agreement between the experiment and calculation is not achieved, elemental features of the observed angular dependence of the MR can be qualitatively explained by the calculation. Thus, we assume that the angular dependence of the magneto-transport properties observed in the normal metallic phase is mainly derived from the geometrical effect of the Fermi surface.