Superconductivity in the Ferromagnet URhGe under uniaxial pressure

TL;DR

Applying uniaxial pressure in URhGe significantly enhances its superconductivity by stimulating triplet pairing mechanisms and affecting magnetic transitions, revealing new insights into pressure-induced superconductivity in ferromagnetic heavy fermion compounds.

Contribution

This study demonstrates that uniaxial stress along the b-axis enhances superconductivity and modifies magnetic transitions in URhGe, highlighting new mechanisms for pressure-induced superconductivity.

Findings

Uniaxial pressure strongly stimulates superconductivity in URhGe.

Stress affects the reentrant superconductivity under magnetic field.

Uniaxial stress lowers the critical field for the transition to the paramagnetic state.

Abstract

Uniaxial pressure applied in the b crystallographic direction perpendicular to spontaneous magnetization in heavy fermion ferromagnet URhGe strongly stimulates superconductivity in this compound. The phenomenological approach allows point out two mechanisms of superconducting temperature raising. They originates from stimulation by the uniaxial stress both intraband and interband amplitudes of triplet Cooper pairing. The phenomenon of reentrant superconductivity under magnetic field along b-axis is also strongly sensitive to the uniaxial stress in the same direction. The uniaxial stress accelerates suppression the Curie temperature by the transversal magnetic field. The emergence of the first order transition to the paramagnetic state occurs at much lower field than in the absence of uniaxial stress.

Figures

b looks similar to the upper critical field temperature dependence $H_{c2}^{b}(T)$ in the other ferromagnetic superconductor UCoGe. Unlike to URhGe the hydrostatic pressure applied to UCoGe suppresses ferromagnetism and stimulate superconductivity [Aoki14 ](#bib.bib1). This allows us to speculate that in the case of UCoGe an uniaxial pressure applied along b-axis can transform $S$-shape $H_{c2}^{b}(T)$ curve in two separate superconducting and reentrant superconducting regions like it is in URhGe in the absence of uniaxial pressure.

Equations

$$F=F_M+F_{el}+F_{Mel},$$

$$F_M={\alpha }_z{M}_z^2+{\beta }_z{M}_z^4+{\delta }_z{M}_z^6+{\alpha }_y{M}_y^2+{\alpha }_x{M}_x^2$$

$$+{\beta }_{xy}{M}_x^2{M}_y^2+{\beta }_{yz}{M}_z^2{M}_y^2+{\beta }_{xz}{M}_z^2{M}_x^2-\mathbf{M}\mathbf{H},$$

$${\alpha }_z={\alpha }_{z0}(T-T_{c0}),{\alpha }_x>0,{\alpha }_y>0.$$

$$F_{el}=\frac{1}{2}{\lambda }_x{u}_{xx}^2+\frac{1}{2}{\lambda }_y{u}_{yy}^2+\frac{1}{2}{\lambda }_z{u}_{zz}^2$$

$$+{\lambda }_{xy}{u}_{xx}{u}_{yy}+{\lambda }_{xz}{u}_{xx}{u}_{zz}+{\lambda }_{yz}{u}_{yy}{u}_{zz}$$

$$+\frac{1}{2}{\mu }_{xy}{u}_{xy}^2+\frac{1}{2}{\mu }_{xz}{u}_{xz}^2+\frac{1}{2}{\mu }_{yz}{u}_{yz}^2,$$

$$F_{Mel}=(p_x{u}_{xx}+p_y{u}_{yy}+p_z{u}_{zz})M_z^2$$

$$+(q_x{u}_{xx}+q_y{u}_{yy}+q_z{u}_{zz})M_y^2+r{u}_{yz}{M}_y{M}_z.$$

$$\frac{\partial (F_{el}+F_{Mel})}{\partial u_{xx}}=0,\frac{\partial (F_{el}+F_{Mel})}{\partial u_{zz}}=0,$$

$$\frac{\partial (F_{el}+F_{Mel})}{\partial u_{yz}}=0\frac{\partial (F_{el}+F_{Mel})}{\partial u_{yy}}=-P_y$$

$$\frac{\partial (F_{el}+F_{Mel})}{\partial u_{xx}}=-P,\frac{\partial (F_{el}+F_{Mel})}{\partial u_{zz}}=-P,$$

$$\frac{\partial (F_{el}+F_{Mel})}{\partial u_{yz}}=0\frac{\partial (F_{el}+F_{Mel})}{\partial u_{yy}}=-P$$

$$F_M=({\alpha }_z+A_z{P}_y)M_z^2+{\beta }_z{M}_z^4+{\delta }_z{M}_z^6$$

$$+({\alpha }_y+A_y{P}_y)M_y^2+{\beta }_{yz}{M}_z^2{M}_y^2-M_y{H}_y.$$

$${\alpha }_z(P)={\alpha }_{z0}(T-T_{c0})+A_z{P}_y,$$

$$T_{c0}(P_y)=T_{c0}-\frac{A_z{P}_y}{{\alpha }_{z0}}.$$

$${\alpha }_y(P_y)={\alpha }_y+A_y{P}_y.$$

$$M_y=\frac{H_y}{2[{\alpha }_y(P_y)+{\beta }_{yz}{M}_z^2]}.$$

$${\alpha }_y(P_y)={\alpha }_y-|A_y|P_y.$$

$${\alpha }_z(P)={\alpha }_{z0}(T-T_{c0})+A_z^{h}P,$$

$${\alpha }_y(P)={\alpha }_y+A_y^{h}P.$$

$$A_z^h<0,A_y^h>0$$

$${\Delta }^{\uparrow }(\mathbf{k},\mathbf{r})={\hat{k}}_x{\eta }_{\uparrow }(\mathbf{r}),$$

$${\Delta }^{\downarrow }(\mathbf{k},\mathbf{r})={\hat{k}}_x{\eta }_{\downarrow }(\mathbf{r})$$

$$T=\epsilon exp\left(-\frac{1}{g}\right),$$

$$g=\frac{g_{1x}^{\uparrow }+g_{1x}^{\downarrow }}{2}+\sqrt{\frac{(g_{1x}^{\uparrow }-g_{1x}^{\downarrow }{)}^2}{4}+g_{2x}^{\uparrow }{g}_{2x}^{\downarrow }}$$

$$g_{1x}^{\uparrow }\propto \frac{⟨{\hat{k}}_x^2{N}_0^{\uparrow }⟩}{(2{\beta }_z{M}_z^2+{\gamma }^z{k}_F^2{)}^2},$$

$$g_{1x}^{\downarrow }\propto \frac{⟨{\hat{k}}_x^2{N}_0^{\downarrow }⟩}{(2{\beta }_z{M}_z^2+{\gamma }^z{k}_F^2{)}^2}.$$

$$2{\beta }_z{M}_z^2={\alpha }_{z0}(T_{c0}(P_y)-T).$$

$$2{\beta }_z{M}_z^2\approx {\alpha }_{z0}(T_{c0}-A_z{P}_y).$$

$$g_{2x}^{\uparrow }\propto ⟨{\hat{k}}_x^2{N}_0^{\uparrow }⟩\left\{\frac{{\gamma }_{xx}^x}{({\alpha }_x+{\beta }_{xz}{M}_z^2+2{\gamma }^x{k}_F^2{)}^2}-\frac{{\gamma }_{yy}^x}{({\alpha }_y(P)+{\beta }_{yz}{M}_z^2+2{\gamma }^y{k}_F^2{)}^2}\right\}\approx -\frac{{\gamma }_{yy}^x⟨{\hat{k}}_x^2{N}_0^{\uparrow }⟩}{({\alpha }_y(P)+{\beta }_{yz}{M}_z^2+2{\gamma }^y{k}_F^2{)}^2},$$

$$g_{2x}^{\downarrow }\propto ⟨{\hat{k}}_x^2{N}_0^{\downarrow }⟩\left\{\frac{{\gamma }_{xx}^x}{({\alpha }_x+{\beta }_{xz}{M}_z^2+2{\gamma }^x{k}_F^2{)}^2}-\frac{{\gamma }_{yy}^x}{({\alpha }_y(P_y)+{\beta }_{yz}{M}_z^2+2{\gamma }^y{k}_F^2{)}^2}\right\}\approx -\frac{{\gamma }_{yy}^x⟨{\hat{k}}_x^2{N}_0^{\downarrow }⟩}{({\alpha }_y(P)+{\beta }_{yz}{M}_z^2+2{\gamma }^y{k}_F^2{)}^2},$$

$$F_M={\alpha }_z(P_y)M_z^2+{\beta }_z{M}_z^4+{\delta }_z{M}_z^6-\frac{1}{4}\frac{H_y^2}{{\alpha }_y(P_y)+{\beta }_{yz}{M}_z^2},$$

$$F_M=-\frac{H_y^2}{4{\alpha }_y(P_y)}+{\tilde{\alpha }}_z{M}_z^2+{\tilde{\beta }}_z{M}_z^4+{\tilde{\delta }}_z{M}_z^6+\dots ,$$

$${\tilde{\alpha }}_z={\alpha }_{z0}(T-T_{c0}(P_y))+\frac{{\beta }_{yz}{H}_y^2}{4({\alpha }_y(P_y){)}^2},$$

Full text

I Introduction

The coexistence of superconductivity and ferromagnetism is the hallmark of heavy fermion uranium compounds UGe2, URhGe and UCoGe (see the recent experimental Aoki14 and theoretical Mineev2016 reviews and references therein). The emergence of the superconducting state at temperatures far below the Curie temperature and very high upper critical field strongly indicate on the spin-triplet Cooper pairing in these materials. One of the most peculiar observations is the phenomenon of reentrant superconductivity in URhGe Levy2005 which is an orthorhombic ferromagnet with spontaneous magnetization along c-axis. At low enough temperature the magnetic field about 1.3 Tesla directed along the $b$-axis suppresses the superconducting state Hardy2005 but at much higher field about 10 Tesla the superconductivity is recreated and exists till the field about 13 Tesla Levy2005 . The maximum of the superconducting critical temperature in this field interval is $\approx 0.4~{}K$. In the same field interval the material transfers from the ferromagnet to the paramagnet state by means of the first-order type transition. The superconducting state exists not only inside of the ferromagnetic state but also in the paramagnetic state separated from the ferromagnetic state by the phase transition of the first order. The theoretical treatment of this phenomenon has been proposed in Ref.5.

There was shown experimentally that a hydrostatic pressure applied to URhGe crystals stimulates ferromagnetism causing an increase of the Curie temperature $T_{c}(P)$ and, at the same time, suppresses the superconducting state decreasing the critical temperature of superconducting phase transition $T_{sc}(P)$ Hardy , as well as the maximum of the superconducting critical temperature of the reentrant superconducting state Miyake2009 . The latter is also shifted to a bit higher field interval. Quite the opposite behavior has been registered recently Braithwaite2016 under uniaxial stress $P_{y}$ in b-direction. In what follows we shall use the $x,y,z$ coordinate axes pinned correspondingly to the $a,b,c$ crystallographic directions. Namely, the uniaxial stress suppresses the ferromagnetism decreasing the Curie temperature $T_{c}(P_{y})$ and stimulate the superconducting state such that the temperature of superconducting transition increases so strongly that leads to the coalescence the superconducting and reentrant superconducting regions in $(H_{y},T)$ phase diagram already at quite moderate uniaxial stress values. The comparison of $(H_{y},T)$ phase diagrams at ambient pressure and at some uniaxial stress is presented in Fig.1.

Here, I show that the stimulation of superconducting state originates from two mechanisms: the Curie temperature suppression stimulating the intraband pairing and the increase of the magnetic susceptibility along b-direction stimulating interband pairing. Then, making use the phenomenological approach developed in Re.5, I consider the $(H_{y},T)$ phase diagram modification caused by uniaxial stress along b-direction. It is demonstrated that the uniaxial stress strongly accelerates the process of Curie temperature suppression by magnetic field along b direction. Also the field induced transformation of the second to the first order ferro-para phase transition occurs at much lower field $H_{y}^{cr}(P_{y})$ values than in a stress absence. This leads to the coalescence of the superconducting and reentrant superconducting states.

II Free energy

The Landau free energy density of an orthorhombic ferromagnet in magnetic field ${\bf H}({\bf r})$ under an external pressure consists of magnetic, elastic and magneto-elastic parts

[TABLE]

where in the magnetic part Mineev2016

[TABLE]

we bear in mind the orthorhombic anisotropy and also the term of the sixth order in powers of $M_{z}$. Here, $x,y,z$ are the coordinates pinned to the $a,b,c$ crystallographic directions correspondingly,

[TABLE]

The elastic part of free energy in an orthorhombic crystal is TheorEl

[TABLE]

where $u_{xx},u_{yy},u_{zz},u_{xy},u_{xz},u_{yz}$ are the components of deformation tensor.

In constant magnetic field ${\bf H}=H_{y}\hat{y}$ the $x$-projection of magnetization $M_{x}=0$, hence, one can take into account just $y$ and $z$ magneto-elastic terms

[TABLE]

To find the deformaition arising under infuence of uniaxial stress $P_{y}$ applied along $b$-axis one must solve the linear equations

[TABLE]

in respect of components of deformation tensor. Whereas in the case of hydrostatic pressure $P$ the corresponding system of equations is

[TABLE]

Solving the equations (6) and substituting the solution back to the sum $F_{el}+F_{Mel}$ given by equations (4) and (5) we obtain the magnetic field $H_{y}$ and the uniaxial pressure $P_{y}$ dependent magnetic part of free energy density

[TABLE]

The coefficients $\beta_{z}$ and $\beta_{yz}$ in this expression differ from corresponding coefficients in Eq. (2). However, this difference is pressure independent so long the magneto-elastic coupling has the simple form given by eq.(5). Hence, in what follows, we keep for these coefficients the same notations as in Eq.(2).

The coefficient at $M_{z}^{2}$ now is

[TABLE]

and the Curie temperature acquires the pressure dependence

[TABLE]

The expression of coefficients $A_{z}$ and $A_{y}$ through elastic and magneto-elastic moduli (see Appendix) don’t give up hopes to get a theoretical estimation of their value. But in fact this is not necessary. As we shall see the important role play the sign of them which is determined experimentally. The Curie temperature decreases with uniaxial pressure Braithwaite2016 what corresponds to positive $A_{z}$ coefficient.

The coefficient at $M_{y}^{2}$ also acquires the pressure dependence

[TABLE]

The equilibrium magnetization projection along the $y$ direction is obtained by minimization of free energy (8) in respect of $M_{y}$

[TABLE]

The measurements Braithwaite2016 shows that the susceptibility along $y$ direction $\chi_{y}(P_{y})=\frac{M_{y}}{H_{y}}$ increases with pressure enhancement. This is owing to both $T_{c0}(P_{y})$ and much stronger $\alpha(P_{y})$ decrease with uniaxial pressure. So, the coefficient $A_{y}$ is proved to be negative, thus

[TABLE]

We shall see in the next Section that both dependencies (10) and (13) cause the enhancement of superconducting critical temperature. Then I will demonstrate that b-direction magnetic softening makes easier the suppression of ferromagnetic state by magnetic field in $y$ direction and stimulates the emergency of reentrant superconducting state.

The hydrostatic pressure creates the coefficients dependencies of the same form

[TABLE]

This case, however,

[TABLE]

what corresponds, as we shall see, to stimulation of ferromagnetism and to suppression of superconductivity reported in Hardy ; Miyake2009 .

III Superconducting critical temperature

The superconducting state in two band (spin-up and spin-down) orthorhombic ferromagnet is described in its simplest form in terms of two complex order parameter amplitudes Mineev2011 ,Mineev2016

[TABLE]

depending on the Cooper pair centre of gravity coordinate ${\bf r}$ and the momentum ${\bf k}$ of pairing electrons. This particular order parameter structure allows to explain the specific temperature dependence of the upper critical field anisotropy in URhGe Hardy2005 , Mineev2016 .

The corresponding critical temperature of transition to the superconducting state is determined by the BCS-type formula

[TABLE]

where the constant of interaction

[TABLE]

is expressed through the constants of intra-band pairing $g_{1x}^{\uparrow},~{}g_{1x}^{\downarrow}$ and the constants of inter band pairing $g_{2x}^{\uparrow},~{}g_{2x}^{\downarrow}$. They are functions of temperature, pressure and magnetic field. Thereby the formula (18) is, in fact, an equation for the determination of the critical temperature of the transition to the superconducting state.

The constants of intra-band pairing interaction for spin-up and spin-down bands are proportional to the averaged over the Fermi surface density of states and the odd-part of susceptibility along the direction of spontaneous magnetization ( see Mineev2016 Eq.(103))

[TABLE]

The magnetization below the Curie temperature is given by

[TABLE]

Assuming that at temperatures far below from $T_{c0}(P_{y})$ this formula is still qualitatively valid we obtain using Eq.(10)

[TABLE]

Thus, the magnetization decreases with uniaxial pressure, what causes in its turn the increase of superconducting interaction constant.

On the other hand, the constants of inter-band pairing interaction are determined by the difference of the odd parts susceptibilities in $x$ and $y$ directions (see Mineev2016 Eq.(104))

[TABLE]

where we have used the smallness of susceptibility along $x$-direction in respect of susceptibility along $y$-axis Hardy2011 . According to Eqs. (13) and (23) the denominators in these expressions are decreasing functions of pressure. Thus, the absolute values of the constant of interband pairing increase with uniaxial pressure.

Thus, all the terms in Eq.(19) increase with the uniaxial pressure increase, what results in the growing up the temperature of superconducting transition shown in Fig.1.

IV Phase transition

in magnetic field perpendicular to easy magnetization axis

The reentrant superconducting state in URhGe arises in high magnetic field along b-axis in vicinity of the first order transition from the ferromagnetic to the paramagnetic state. The change of the phase transition type from the second to the first order has been described phenomenologically in Ref.5. With aim to establish the uniaxial pressure dependence of the transtion transformation we reproduce this derivation.

The equilibrium magnetization projection along the $y$ direction is given by (12). Substituting this formula to Eq.(8) we obtain

[TABLE]

that gives after expansion of the denominator in the last term,

[TABLE]

where

[TABLE]

We see that under a magnetic field perpendicular to the direction of spontaneous magnetization the Curie temperature decreases as

[TABLE]

Thus, at a finite field $H_{y}$ the Curie temperature suppression by uniaxial pressure occurs much faster than in the field absence.

The coefficient $\tilde{\beta}_{z}$ also decreases with $H_{y}$ and reaches zero at

[TABLE]

At this field under fulfillment the condition,

[TABLE]

the Curie temperature (31) is still positive and at

[TABLE]

phase transition from a paramagnetic to a ferromagnetic state becomes the transition of the first order (see Fig.9 in the Ref.2). The point $(H_{y}^{cr},T_{c}(H_{y}^{cr}))$ on the line paramagnet-ferromagnet phase transition is a tricritical point. The pressure dependence

[TABLE]

roughly corresponds to the observed experimentally Braithwaite2016 pressure dependence $H_{R}(P_{y})$. An uniaxial pressure enhancement decreases the field of the first order phase transition from ferromagnetic to paramagnetic state.

The minimization of the free energy Eq. (27) gives the value of the order parameter in the ferromagnetic state,

[TABLE]

The minimization of the free energy in the paramagnetic state,

[TABLE]

in respect $M_{y}$ gives the equilibrium value of magnetization projection on axis $y$ in paramagnetic state,

[TABLE]

Substitution back in Eq. (36) yields the equilibrium value of free energy in the paramagnetic state,

[TABLE]

On the line of the phase transition of the first order from the paramagnetic to ferromagnetic state determined by the equations StatPhysI

[TABLE]

the order parameter $M_{z}$ has the jump (see Fig.10 in the Ref.2) from

[TABLE]

in the ferromagnetic state to zero in the paramagnetic state. Its substitution back in equation $F_{M}=F_{para}$ gives the equation of the first-order transition line,

[TABLE]

that is

[TABLE]

The corresponding jump of $M_{y}$ (see Fig.10 in Ref.1) is given by

[TABLE]

V Concluding remarks

We have shown that both the Curie and the superconducting critical temperature are changed with the pressure applied to URhGe specimen. The functional pressure dependences can be established from general phenomenological considerations. Whereas the direction of changes must be chosen by comparison with the experimental findings according to which uniaxial in b-direction and hydrostatic pressure act in the opposite sense: the first suppresses the ferromagnetism and enhances superconductivity, the second stimulates ferromagnetism and suppresses superconducting state. The phenomenological approach allows to point out two mechanisms of superconducting temperature raising. They originates from stimulation by the uniaxial stress both intraband and interband amplitudes of triplet Coper pairing.

The magnetic field in b-direction decreases the Curie temperature and leads to the transformation of the ferro-para phase transition from the second to the first order. The pairing interaction in vicinity of the first order transition from ferromagnetic to paramagnetic state caused by field $H_{y}$ is strongly increased in comparison with its zero field value. This effect proves to be stronger than the orbital suppression of superconductivity by magnetic field and leads to the re-appearance of superconducting state at field of the order 10 Tesla. Here we have demonstrated that in presence of an uniaxial pressure along b-axis the process of the ferromagnetism suppression by magnetic field occurs much faster. The field induced transformation of the second to the first order ferro-para phase transition occurs at much lower field $H_{y}^{cr}(P_{y})$ values. The effect is so strong that even small uniaxial stress causes the coalescence Braithwaite2016 of superconducting and the re-entrantant superconducting area in the $(H_{y},T)$ phase diagram shown in Fig.1.

The shape of superconducting region drown in Fig.1b looks similar to the upper critical field temperature dependence $H_{c2}^{b}(T)$ in the other ferromagnetic superconductor UCoGe. Unlike to URhGe the hydrostatic pressure applied to UCoGe suppresses ferromagnetism and stimulate superconductivity Aoki14 . This allows us to speculate that in the case of UCoGe an uniaxial pressure applied along b-axis can transform $S$-shape $H_{c2}^{b}(T)$ curve in two separate superconducting and reentrant superconducting regions like it is in URhGe in the absence of uniaxial pressure.

Acknowledgements.

I am indebted to D.Braithwaite and J.-P. Brison for the helpful discussions of their recent experimental results.

Appendix A

The explicit expressions for $A_{z}$ and $A_{y}$ coefficients through the elastic and magneto-elastic moduli are

[TABLE]

[TABLE]

and

[TABLE]