Metallic quantum ferromagnets

An overview of quantum phase transitions (QPTs) in metallic ferromagnets, discussing both experimental and theoretical aspects, is given. These QPTs can be classified with respect to the presence and strength of quenched disorder: Clean systems generically show a discontinuous, or first-order, QPT from a ferromagnetic to a paramagnetic state as a function of some control parameter, as predicted by theory. Disordered systems are much more complicated, depending on the disorder strength and the distance from the QPT. In many disordered materials the QPT is continuous, or second order, and Griffiths-phase effects coexist with QPT singularities near the transition. In other systems the transition from the ferromagnetic state at low temperatures is to a different type of long-range order, such as an antiferromagnetic or a spin-density-wave state. In still other materials a transition to a state with glasslike spin dynamics is suspected. The review provides a comprehensive discussion of the current understanding of these various transitions and of the relation between experiment and theory.

Figure 1

Observed phase diagram of ${\mathrm{UGe}}_2$ in the space spanned by temperature ($T$), pressure ($P$), and magnetic field ($H$). Solid red curves represent lines of second-order transitions, and blue planes represent first-order transitions. Also shown are the tricritical point (TCP), and the extrapolated “quantum-critical end points” (QCEP) at the wing tips. From [292].

Figure 2

Schematic phase diagrams observed in ferromagnetic (FM) systems that show, at the lowest temperatures realized, (a) a discontinuous transition and tricritical wings in a magnetic field, (b) a continuous transition, (c) a change to spin-density-wave (SDW) or antiferromagnetic (AFM) order, and (d) a continuous transition in strongly disordered systems that may be accompanied by quantum Griffiths effects or spin-glass freezing in the tail of the phase diagram.

Figure 3

Phase diagram of MnSi. In the temperature-pressure ($T\text{−}p$) plane the transition temperature drops from $T_{\mathrm{C}}=29.5\mathrm{K}$ at ambient pressure and changes from second to first order at $p^{*}=12\mathrm{kbar}$, where $T_{\mathrm{C}}\approx 12\mathrm{K}$. $T_{\mathrm{C}}$ vanishes at $p_{\mathrm{c}}=14.6\mathrm{kbar}$. In the magnetic field, pressure ($B\text{−}p$) plane at $T=0$, and everywhere across the shaded wing, the transition is first order up to a “critical end point” (see footnote ) estimated to be located at $B_{\mathrm{m}}=0.6\mathrm{T}$ and $p_{\mathrm{m}}=17\mathrm{kbar}$. From [399].

Figure 4

$\mu \mathrm{SR}$ results for the volume fraction with static magnetic order. The nonzero volume fraction less than unity at $T=0$ for intermediate pressures indicates phase separation, which in turn is indicative of a first-order transition. From [514].

Figure 5

Phase diagram of ${\mathrm{UGe}}_2$ in the temperature-pressure plane. Shown are the paramagnetic (PM) phase, two ferromagnetic phases (FM1 and FM2), and the tricritical point (TCP). The critical point marked CEP (see footnote ) is related to the transition between the phases FM1 and FM2. From [502].

Figure 6

Phase diagram of URhGe in the space spanned by temperature and magnetic fields in the $b$ and $c$ directions. The dark shaded regions indicate the presence of superconductivity. From [216].

Figure 7

Phase diagram of UCoGe in the temperature-pressure plane, showing the ferromagnetic and superconducting phases. The magnetic transition is first order for all pressure values ([195]). From [468].

Figure 8

Curie temperature of URhGe doped with Co, Si, or Rh as a function of the dopant concentration. The transition in pure URhGe is second order, and in pure UCoGe, first order. From [431].

Figure 9

(a) Magnetostriction measured along the $c$ axis with $H\parallel c$ for temperatures between 2 and 21 K (every 1 K). (b) Temperature vs field evolution of the metamagnetic transition which changes from first order to a crossover for $T>T_0=11\mathrm{K}$. From [17].

Figure 10

$T\text{−}P\text{−}H$ phase diagram of UCoAl. The tricritical wings are determined by the observation of a first-order metamagnetic transition at $H_m$ (red dots); they are bounded by lines of second-order transitions at $T_0$ and end in quantum “critical end points” (QCEPs) (see the explanation in the Fig. 1 caption). The critical pressure $P_{\mathrm{c}}$ is negative and the tricritical point (TCP) is not accessible. From [17].

Figure 11

Upper panel: Temperature-pressure phase diagram of URhAl in zero field determined from resistivity measurements. Lower panel: Temperature-pressure-field diagram inferred from metamagnetic behavior observed in an external field. From [461].

Figure 12

Schematic phase diagram of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ in the space spanned by temperature ($T$), hydrostatic pressure ($P$), and magnetic field ($H$), with $H\perp c$. Ambient pressure ($P_0$) is indicated by the solid purple line. The tricritical point (TCP) is not accessible, but the tricritical wings are observed by sweeping the magnetic field at fixed temperature [dashed purple lines and inset (i)]. The critical temperature $T^{*}$ can be tuned by rotating the magnetic field by an angle $\theta$ out of the magnetically easy $ab$ plane. As $T^{*}$ decreases, the wing tips split and a phase displaying a strong transport anisotropy is found in between two sheets, with a second-order phase boundary on top [inset (ii) and Fig. 13]. From [543].

Figure 13

$T\text{−}H$ phase diagram of ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ at ambient pressure with $H\parallel c$. The near-vertical lines are the two sheets of the tricritical wings, with the novel phase in between. The transition below (above) the red arrows is first (second) order. From [180].

Figure 14

Observed phase diagram of ${\mathrm{Ni}}_x{\mathrm{Pd}}_{1-x}$. Solid dots are data taken by [376], and the other symbols represent earlier data. From [376].

Figure 15

Hertz-type scaling behavior as observed in ${\mathrm{Ni}}_3{\mathrm{Al}}_{1-x}{\mathrm{Ga}}_x$. From left to right: $T_{\mathrm{C}}$ vs $x$ phase diagram, inverse magnetic susceptibility vs $T^{4/3}$ for $x=x_{\mathrm{c}}$, and magnetization squared vs $T^{4/3}$ for various $x$. From [551].

Figure 16

Temperature-pressure ($T\text{−}P$) phase diagram of ${\mathrm{UNiSi}}_2$ from resistivity, ac susceptibility, and specific-heat measurements. The small, shaded (turquoise) region represents a separate phase characterized by weak FM and a large Sommerfeld coefficient. In the PM phase the Sommerfeld coefficient remains large as does the resistivity. From [467].

Figure 17

Temperature dependence of the ac susceptibility ${\chi }^{\prime }(T)$ measured with a modulation-field amplitude ${\mu }_{0}H=15\mu \mathrm{T}$ parallel and perpendicular, respectively, to the $c$ axis. The large value ${\chi }^{\prime }(T_{\mathrm{C}})\approx 200\times {10}^{-6}{\mathrm{m}}^3/\mathrm{mol}$ indicates a FM phase transition. The dashed line indicates that ${\chi }_{\parallel }^{\prime }(T)\propto T^{-2/3}$ above $T_{\mathrm{C}}$. From [476].

Figure 18

(a) Phase diagram of ${\mathrm{YbNi}}_4({\mathrm{P}}_{1-x}{\mathrm{As}}_x{)}_2$. (b) Specific-heat (blue, left axis) and volume thermal expansion coefficient (green, right axis) for the sample with $T_{\mathrm{C}}\approx 25\mathrm{mK}$ ($x=0.08$). Inset: $T$ dependence of the Grüneisen ratio $\mathrm{\Gamma }(T)=\beta (T)/C(T)\propto T^{-0.22}$. From [476].

Figure 19

Phase diagram of ${\mathrm{Nb}}_{1-y}{\mathrm{Fe}}_{2+y}$. From [363].

Figure 20

Magnetization isotherms at 2 K with $H\parallel c$ and $H\perp c$. From [296].

Figure 21

Pressure-temperature-magnetic field phase diagram of CeRuPO derived from resistivity measurements. At ambient pressure and $H=0$ the ground state is FM, but changes into an AFM one at about 0.7 GPa. Magnetism is suppressed at $p_{\mathrm{c}}\approx 2.8\mathrm{GPa}$. The AFM state is identified by the transition temperature decreasing with increasing field (solid black lines). At even larger fields ($H_m$) a metamagnetic crossover occurs from a PM state to a polarized PM (PPM) state. $T^{*}$ indicated the Kondo coherence temperature which increases with increasing pressure. From [293].

Figure 22

$T\text{−}x$ phase diagram of ${\mathrm{CeFeAs}}_{1-x}{\mathrm{P}}_x\mathrm{O}$ obtained by a variety of techniques (list in the right inset) for single and polycrystalline samples. Red and blue dotted lines indicate the AFM ordering temperature of the Fe and Ce sublattices, respectively. Superconductivity (SC) is found near $x=0.3$. The dotted black line is the Curie temperature for the FM order of the Ce atoms. From [238], and [240].

Figure 23

Temperature dependence of the in-plane spin fluctuations $S_{\perp }=(1/2T_1{)}_{H\parallel c}$ at ${\mu }_{0}H=0.5\mathrm{T}$ for various values of $x$. The peak indicates the FM transition temperature $T_{\mathrm{C}}$, which is visible only for $x\le 0.85$. For $x>0.85$, $S_{\perp }$ is constant at low $T$. The inset shows the $x$ dependence of $S_{\perp }$ at a temperature of 200 mK. $S_{\perp }$ peaks at $x_{\mathrm{c}}$ indicating the presence of a QCP. From [277].

Figure 24

Lower panel: $x\text{−}T$ phase diagram of ${\mathrm{CeRu}}_{1-x}{\mathrm{Fe}}_x\mathrm{PO}$ derived from NMR studies. Strong fluctuations are indicative of a QCP at $x_{\mathrm{c}}\approx 0.86$. At larger $x$ the ground state is a heavy-fermion (HF) paramagnet, bounded by the temperature $T_{\max }$ where the Knight shift shows a peak. Upper panel: The exponent $n$ is indicative of the effective dimensionality of magnetic correlations, with $n=1.5$ corresponding to 2D correlations. From [278].

Figure 25

Isothermal magnetization of a single crystal of ${\mathrm{CeRu}}_{0.22}{\mathrm{Fe}}_{0.78}\mathrm{PO}$ for $H\parallel c$ and $H\perp c$ at $T\approx 1.8\mathrm{K}$. Note the absence of a remanent magnetization at $H=0$. The inset shows a weak hysteresis loop near ${\mu }_{0}H=1.1\mathrm{T}$ for $H\parallel c$. From [239].

Figure 26

Pressure-temperature phase diagram of ${\mathrm{CeAgSb}}_2$. $T_{\mathrm{C}}$ and $T_{\mathrm{N}}$ were determined from $d\rho /dT$ while $T_{\text{mag}}$ is from ac-calorimetry measurements. From [465].

Figure 27

(a) $T\text{−}x$ phase diagram of $\mathrm{Yb}({\mathrm{Rh}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{Si}}_2$ as measured by [283] showing the transition temperatures $T_L$ and $T_{\mathrm{N}}$. The red arrow marks the sample with $x=0.27$ which is FM. (b) Real part ${\chi }^{\prime }(T)$ of the susceptibility for $\mathrm{Yb}({\mathrm{Rh}}_{0.73}{\mathrm{Co}}_{0.27}{)}_2{\mathrm{Si}}_2$ in different magnetic fields with $B\parallel c$. The sharp peak at $T_{\mathrm{C}}=1.30\mathrm{K}$ and $B=0$ is suppressed and shifted toward higher $T$ with increasing field. (c) Temperature dependence of the imaginary part ${\chi }^{\prime \prime }(T)$ of the susceptibility. From [312].

Figure 28

Magnetic phase diagram of ${\mathrm{CePd}}_{1-x}{\mathrm{Rh}}_x$. Left scale: Composition dependence of the ordering (freezing) temperature $T_{\mathrm{C}}$ ($T_{\mathrm{C}}^{*}$) deduced from various measurement techniques: magnetization ($M$), ac susceptibility (${\chi }_{\mathrm{a}\mathrm{c}}^{\prime }$), thermal expansion ($\beta$) and specific heat ($C$). The inset shows $T_{\mathrm{C}}$ ($x$) values observed in ${\chi }_{\mathrm{a}\mathrm{c}}^{\prime }(T)$ for $x>0.7$ in polycrystals and single crystals. Right scale: The Weiss temperature ${\theta }_{\mathrm{P}}$. From [453] and [533].

Figure 29

Comparison of thermodynamic data for ${\mathrm{CePd}}_{1-x}{\mathrm{Rh}}_x$ with the quantum-Griffiths-phase scenario (cf. Sec. 3d). Main figure: Field dependence of the magnetization $M$ for a single crystal with $x=0.8$ at 50 mK. The data follow a power law $M\propto H^{\lambda }$ with $\lambda =0.21$. Inset (a): $4f$ Sommerfeld coefficient for polycrystals with $x=0.87$ and 0.9 ([408]) plotted on a double-logarithmic scale. Solid lines indicate a $T^{\lambda -1}$ power-law behavior. Inset (b): $T$ dependence of the ac susceptibility ${\chi }^{\prime }(T)$ of three polycrystals with $x=0.8$, 0.85, and 0.87. The lines are fits to a $T^{\lambda -1}$ power law. From [533].

Figure 30

$x\text{−}T$ phase diagram of ${\mathrm{Ni}}_{1-x}{\mathrm{V}}_x$. $T_{\max }$, where the ac susceptibility peaks, marks the freezing into the cluster-glass (CG) state. For $0.114\le x\le 0.15$ effects consistent with a quantum Griffiths phase (GP) were observed in the susceptibility ($\chi \propto T^{-\gamma }$) and magnetization ($M\propto H^{\alpha }$), see Fig. 31. The orange squares mark the crossover from the GP to the CG phase. The inset shows the $x$ dependence of the exponents. From [446].

Figure 31

The low-field susceptibility ${\chi }_m=M/H-{\chi }_{\text{orb}}$ vs $T$, and the magnetization $M_m=M-{\chi }_{\text{orb}}H$ at 2 K vs $H$ for samples with $0.114\le x\le 0.15$. ${\chi }_{\text{orb}}$ is the orbital contribution to $\chi$. The dashed lines indicate power-law behaviors ${\chi }_m\propto T^{-\gamma }$ and $M_m\propto H^{\alpha }$. From [513].

Figure 32

$x\text{−}T$ phase diagram of ${\mathrm{UNi}}_{1-x}{\mathrm{Co}}_x{\mathrm{Si}}_2$. From [409].

Figure 33

$T\text{−}x$ phase diagram of ${\mathrm{U}}_{1-x}{\mathrm{Th}}_x{\mathrm{NiSi}}_2$ derived from magnetization $M(T)$ and specific-heat $C(T)$ measurements. The circles correspond to the maxima in $\chi (T)=M(T)/B$ or minima in $\partial M(T)/\partial T$, and the triangles correspond to the minima in $\partial [C(T)/T]/\partial T$. The arrow ($x_{\mathrm{c}}$) indicates the position of the putative FM QCP obtained from extrapolating the low-$x$ curvature of the phase separation line. From [407].

Figure 34

$x\text{−}T$ phase diagram of ${\mathrm{CeTi}}_{1-x}{\mathrm{V}}_x{\mathrm{Ge}}_3$. $T_{\mathrm{C}}$ decreases approximately linearly with increasing $x$, as indicated by the dotted line. Also shown is a fit by $T_{\mathrm{C}}\propto (x_{\mathrm{c}}-x{)}^{3/4}$ (solid line), which is expected within the Herz-Millis-Moriya theory (cf. Sec. 3c2). From [281].

Figure 35

Contour plot of the remanent magnetization in the $T\text{−}x$ plane for an epitaxial film of ${\mathrm{Sr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{RuO}}_3$. Black and white symbols indicate the transition temperature determined from susceptibility and magnetization measurements, respectively. From [114].

Figure 36

Normalized muon-spin asymmetry function $G(t,B)$ in CeFePO at different temperatures. The lines are fits to $G(t,B)=G_1{e}^{-({\lambda }_{T}t)}+G_2{e}^{-({\lambda }_{L}t{)}^{\beta }}$. The peak in ${\lambda }_L(T)$ and the strong increase of ${\lambda }_L(T)$ mark the spin-freezing temperature $T_g$. From [311].

Figure 37

Schematic phase diagram in the space spanned by temperature ($T$), hydrostatic pressure ($p$), and magnetic field ($h$). Shown are the FM and PM phases, the tricritical point (TCP), and various QCPs. Solid and dashed lines denote second- and first-order transitions, respectively. The tricritical wings emerging from the TCP are surfaces of first-order transitions. The three panels show the predicted change with increasing disorder. From [433].

Figure 38

Phase diagram in the temperature-disorder plane as given by Eq. (3.35). The dashed line reflects the classical dilution effect of the disorder only [$a=0$ in Eq. (3.35)]; the solid (red) line also reflects the quantum effects, with $a=1$, $b=10$, due to the diffusive dynamics of the electrons.

Figure 39

Proposed phase diagram for a model allowing for spiral and spin-nematic order. $\mu$ is the chemical potential, and $g$ is the exchange coupling. From [254].