Magnetic-field-induced quantum critical point in YbPtIn and $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ single crystals

Detailed anisotropic ($H\Vert ab$ and $H\Vert c$) resistivity and specific heat measurements were performed on on-line-grown YbPtIn and solution-grown $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ single crystals for temperatures down to $0.4\mathrm{K}$ and fields up to $140\mathrm{kG}$; $H\Vert ab$ Hall resistivity was also measured on the $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ system for the same temperature and field ranges. All these measurements indicate that the small change in stoichiometry between the two compounds drastically affects their ordering temperatures ($T_{\mathrm{ord}}\approx 3.4\mathrm{K}$ in YbPtIn and $\sim 2.2\mathrm{K}$ in $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$). Furthermore, a field-induced quantum critical point is apparent in each of these heavy fermion systems, with the corresponding critical field values of $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ (${\mathrm{H}}_c^{ab}$ around $35–45\mathrm{kG}$ and ${\mathrm{H}}_c^c\approx 120\mathrm{kG}$) also reduced compared to the analogous values for YbPtIn (${\mathrm{H}}_c^{ab}\approx 60\mathrm{kG}$ and ${\mathrm{H}}_c^c>140\mathrm{kG}$).

Figure 1

(a) Anisotropic susceptibilities for $H=2\mathrm{kG}$ (with the low-temperature part enlarged in the inset) and (b) field-dependent magnetization at $T=2\mathrm{K}$ for YbPtIn (open symbols) and $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ (solid symbols).

Figure 2

(a) Low-temperature specific-heat and (b) resistivity data for $H=0$, for YbPtIn (open symbols) and $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ (solid symbols), with the resistivity error bars shown for the lowest temperature $(T=0.4\mathrm{K})$; the transition temperatures are indicated by two dotted lines (for YbPtIn) and one dashed line (for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$). Note that the error bars in (b) are geometrical, and thus they scale proportionally at higher temperatures.

Figure 3

Temperature-dependent resistivity for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ and YbPtIn, for $H=0$ (symbols) and $H=140\mathrm{kG}$ (solid lines).

Figure 4

(a) Low-temperature specific-heat curves for YbPtIn, for $H\Vert ab$, with the arrows indicating the peak positions associated with magnetic phase transitions; inset: the $H=0$ $C_P∕T\left(T^2\right)$ curve, with its linear fit below $5\mathrm{K}$ (dotted line) extrapolated down to $T=0$ to provide a rough estimate of the electronic specific heat coefficient $\gamma \simeq 500\mathrm{mJ}∕\mathrm{mol}{\mathrm{K}}^2$ (b) low-temperature part of the $C_m∕T\left(T^2\right)$ curves for various values of the applied field.

Figure 5

Semilog plot of $C_m∕T$ vs $T$ for (a) $H=0–60\mathrm{kG}$ and (b) $H=60–140\mathrm{kG}$. All curves (except for the $H=140\mathrm{kG}$ curve) are shifted up by multiples of $100\mathrm{mJ}∕\mathrm{mol}{\mathrm{K}}^2$. The dotted line (for $H=60\mathrm{kG}$) is a guide to the eye for the largest region of logarithmic divergency of $C_m∕T$.

Figure 6

Low-temperature resistivity data for YbPtIn taken at different applied fields, for $H\Vert ab$; the small arrows indicate the magnetic transition temperatures.

Figure 7

Low-temperature part of the (a) $\rho \left(T\right)$ data for $H=60$, 70, 80, and $100\mathrm{kG}$ and (b) $\rho \left(T^2\right)$ for $H=100$, 120, and $140\mathrm{kG}$.

Figure 8

$\rho \left(H\right)$ isotherms for YbPtIn, for $H\Vert ab$ and $T=0.4–3.6\mathrm{K}$, $T=0.2\mathrm{K}$; the arrows point to the transition fields at the lowest temperature, with the inset exemplifying how these critical fields are determined for $T=0.8\mathrm{K}$.

Figure 9

$H\Vert ab$ $T\text{−}H$ phase diagram for YbPtIn, as determined from resistivity (solid symbols) and specific heat (open symbols) data.

Figure 10

$H\Vert c$ specific heat data for YbPtIn, for $H=0$, 10, 20, 40, 60, and $80\mathrm{kG}$; small arrows indicate the positions of peaks possible associated with magnetic phase transitions (dotted arrows: peaks on $C_P\left(T\right)$ data for possibly torqued sample [see text]).

Figure 11

(a) Low-temperature $\rho \left(T\right)$ data for YbPtIn taken at different applied fields, for $H\Vert c$. (b) $d\rho ∕dT$ curves for $H=20$, 80, 100, 120, and $140\mathrm{kG}$. On both plots, the small arrows indicate the magnetic transition temperatures.

Figure 12

$\rho \left(H\right)$ isotherms for $T=0.4,0.5,1–4\mathrm{K}$ $(\Delta T=0.25\mathrm{K}$) and $5\mathrm{K}$. Small arrow indicates the critical field position at the lowest temperature $(T=0.4\mathrm{K})$.

Figure 13

$H\Vert c$ $T\text{−}H$ phase diagram for YbPtIn, as determined from resistivity (solid symbols) and specific-heat (open symbols) data. Error bars on points from $\rho \left(T\right)$ data at $H=20$, 80, 100, 120, and $140\mathrm{kG}$, and from $\rho \left(H\right)$ at $T=0.4\mathrm{K}$ shown as a caliper of the errors in determining the points on this phase diagram.

Figure 14

(a) Low-temperature specific heat curves for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$, for $H\Vert ab$, with the arrows indicating the positions of the peaks associated with magnetic ordering; inset: the $H=0$ $C_P∕T\left(T^2\right)$ curve, with its linear fit below $5\mathrm{K}$ (dotted line) extrapolated down to $T=0$ to provide a rough estimate of the electronic specific-heat coefficient $\gamma \simeq 500\mathrm{mJ}∕\mathrm{mol}{\mathrm{K}}^2$ (b) low-temperature part of the $C_m∕T\left(T^2\right)$ curves for various values of the applied field.

Figure 15

Semilog plot of $C_m∕T$ vs $T$ for (a) $H=0–40\mathrm{kG}$ and (b) $H=40–140\mathrm{kG}$. All curves (except for the $H=140\mathrm{kG}$ one) are shifted up by multiples of $200\mathrm{mJ}∕\mathrm{mol}{\mathrm{K}}^2$. The dotted lines (for $H=35$ and $40\mathrm{kG}$) are guides to the eye for the linear regions on the $C_m∕T$ curves for $H$ possibly just above and below $H_c^{ab}$ (see text).

Figure 16

$\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ low-temperature resistivity data for $H\Vert ab$, for (a) $H=0$, 10, 20, 25, 30, 35, and $40\mathrm{kG}$ and (b) $H=40$, 42.5, 45, 50, 55, 60, 70, 80, 100, 120, and $140\mathrm{kG}$. Small arrows in (a) indicate the possible ordering temperatures, whereas in (b) the solid lines are guides to the eye for the potentially linear regions on the $H=42.5$, 45, 50, and $55\mathrm{kG}$ curves.

Figure 17

Low-temperature part of the $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ $\rho \left(T^2\right)$ data, for (a) $H=42.5$, 45, 50, 55, 60, and $70\mathrm{kG}$ and (b) $H=70$, 80, 100, 120, and $140\mathrm{kG}$, with solid lines as guides to the eye for the potentially linear regions for the $H⩾45\mathrm{kG}$ curves.

Figure 18

$\rho \left(H\right)$ isotherms for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$, for $H\Vert ab$ and (a) $T=0.4–1.0\mathrm{K}$ $(\Delta T=0.1\mathrm{K})$, $1.2–2.2\mathrm{K}$ $(\Delta T=0.2\mathrm{K})$, and (b) $T=2.2–3.0\mathrm{K}$ $(\Delta T=0.2\mathrm{K})$, 4, 6, 8, and $10\mathrm{K}$; the arrows point to the transition fields at $T=0.4\mathrm{K}$, with the inset exemplifying how these critical fields were determined.

Figure 19

$H\Vert ab$ phase diagram for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$, as determined based on specific-heat (open symbols) and transport (solid symbols) data. The error bars on the high-field points represent the uncertainty in determining the cross over temperatures.

Figure 20

$H\Vert c$ specific heat data for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$, for $H=0$, 20, and $40\mathrm{kG}$; small arrows indicate the positions of peaks possibly associated with the magnetic phase transition.

Figure 21

(a) $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ low-temperature resistivity data for $H\Vert c$, for $H=0–140\mathrm{kG}$ $(\Delta H=10\mathrm{kG})$, with the large dots marking the possible magnetic ordering temperature; inset: $H=0$ $d\rho ∕dT$, with the small arrow marking the transition temperature. (b) $\rho \left(H\right)$ isotherms for $T=0.4–3.0\mathrm{K}$ $(\Delta T=0.1\mathrm{K})$. The critical fields for the possible phase transition were determined as local maxima in $d\rho ∕dH$ (as illustrated in the inset for $T=1.2–1.6\mathrm{K}$) and are shown as large squares.

Figure 22

$T\text{−}H$ phase diagram for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ $(H\Vert c)$, as determined from the specific heat (open symbols) and resistivity (solid symbols) measurements.

Figure 23

(a) Low-temperature part of the $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}\ln \left(\Delta \rho \right)$ vs $\ln T$ data for $H=45$, 50, 55, 60, 70, 80, 100, 120, and $140\mathrm{kG}$, with their respective linear fits at very low $T$ (solid lines). The large triangles indicate the temperatures at which deviations from linearity occur. (b) Exponent $\beta$ (left) and crossover temperature $T_{\max }$ (right) as a function of $H$, with the corresponding error bars.

Figure 24

(a) Low-temperature part of the YbPtIn $\ln \left(\Delta \rho \right)$ vs $\ln T$ data for $H=70$, 80, 100, 120, and $140\mathrm{kG}$, with their respective linear fits at very low $T$ (solid lines). The large triangles indicate the temperatures at which deviations from linearity occur. (b) Exponent $\beta$ (left) and crossover temperature $T_{\max }$ (right) as a function of $H$, with their corresponding error bars.

Figure 25

Revised $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ phase diagram $(H\Vert ab)$. The solid lines correspond to the phase lines from the $T\text{−}H$ phase diagram in Fig. 19, and the symbols delineate the new phase lines as determined from the $\rho \left(T^{\beta }\right)$ data, using various criteria as described in the text.

Figure 26

Revised YbPtIn phase diagram $(H\Vert ab)$. The solid lines correspond to the phase lines from the $T\text{−}H$ phase diagram in Fig. 9, and the symbols delineate the new phase line as determined from the $\rho \left(T^{\beta }\right)$ data, using various criteria as described in the text.

Figure 27

Field-dependent Hall resistivity for $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ $(H\Vert ab)$ for (a) $T=25–300\mathrm{K}$ and (b) $T=0.4–20\mathrm{K}$ (except for the $T=0.4$ and $25\mathrm{K}$, all curves are shifted up by multiples of $0.2\mu \Omega$ for clarity). The lines represent the phase transitions from the phase diagram in Fig. 19; the large diamonds indicate the ${\rho }_{\mathrm{H}}$ minima.

Figure 28

Temperature-dependent Hall coefficient of $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ $(H\Vert ab)$), with low-field (open symbols) and high-field (solid symbols) points determined as described in the text.

Figure 29

Alternative definitions of the Hall coefficient of $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ $(H\parallel ab)$: (a) $R_{\mathrm{H}}={\rho }_{\mathrm{H}}∕H$ and (b) $R_{\mathrm{H}}=d{\rho }_{\mathrm{H}}∕dH$, measured at various temperatures. Except for the $T=0.4\mathrm{K}$ ones, all other curves are shifted for clarity, by multiples of (a) $\mathrm{0.01}n\Omega \mathrm{cm}∕\mathrm{G}$ and (b) $0.2n\Omega \mathrm{cm}∕\mathrm{G}$. The large dots are the ${\rho }_{\mathrm{H}}$ minima shown in Fig. 27, and the dotted line represents the crossover line, determined by the maximum $H$ on the low field linear fits (see text).

Figure 30

Revised $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ phase diagram $(H\parallel ab)$. The solid lines now represent the phase lines from the $T\text{−}H$ phase diagram in Figs. 19, 25, and the symbols delineate the new phase line as determined from the Hall resistivity data, using various criteria as described in the text.

Figure 31

The $H\Vert ab$ field-dependent electronic specific heat coefficient $\gamma$ of YbPtIn (open symbols) and $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ (solid symbols).

Figure 32

The $H\Vert ab$ $T^2$-resistivity coefficient A as a function of field, for (a) YbPtIn and (b) $\mathrm{Yb}{\mathrm{Pt}}_{0.98}\mathrm{In}$ (symbols); the solid line represents a $1∕(H-H_{c0}^{\mathrm{calc}})$ fit, from which the expected critical field values $H_{c0}^{\mathrm{calc}}$ were estimated (see text). Inset: log plot of A vs $\gamma$ (symbols), with a linear fit (solid line) used to estimate the $A∕{\gamma }^2$ ratio.