Superconductivity in Uranium Ferromagnets

V.P.Mineev Commissariat a l’Energie Atomique, Grenoble, FranceLandau Institute for Theoretical Physics, Chernogolovka, Russia

Outline

• Order parameters of ferromagnetic superconductors• Microscopic description• Some physical properties• Reentrant superconductivity in URhGe• Particular problems

Order parametersof ferromagnetic superconductors

Superconductivity and Ferromagnetism in ternary compounds

M.B.Maple, J.Mag.Mag.Mat. 31-34, 479 (1983)

Superconductivity and Ferromagnetism in uraniumcompounds

S.Saxena et al 2000D.Aoki et al 2001N.T.Huy et al 2007

Uranium ferromagnets

Symmetry of normal, ferromagnet and superconducting states

Two band sc ferromagnet

Triplet Pairing in 3He

Superconducting states in orthorhombic fm

PSC→FSC

Unitary state like B-phase

Nonunitary superconducting A-state

Gor’kov equations and quasiparticles spectrum

H

T

mixed

Hint.

Hc2

Equal-spin-pairing state (d0=0)

Goal

Our goal is to find a microscopic modelgenerating these superconducting states

Microscopic approach

Interaction mediatedby bosonic excitations

He-3Eliashberg

Interaction mediated magnetic fluctuations

Isotropic Fermi liquid

Itinerant isotropic Ferromagnet

UGe2 and UCoGe

One must take into account the strong anisotropy and the mixed localized-itinerant nature of ferromagnetism

Properties of susceptibility

Dzyaloshinskii-Moriya

Triplet pairing 1

Triplet pairing 2

underlined terms determines critical temperature in superfluid 3He

The susceptbility is the media susceptibility butnot the electron gaz susceptibility.

The susceptibility is a tensor.The diagonal pairing amplitudes are due to

longitudinal magnetic fluctuations.The off-diagonal pairing amplitudes arise only

due to orthorhombic symmetry explicitlytaken into account. They originate fromspin-orbit interaction.

Triplet pairing 3

Ignoring the interband pairing, that is the terms containing products

Even if we ignore the interband pairing the zero spin part of the order parameter Δ0 still exists. Due tospin-orbit coupling Δ0 is induced by pairing components with with parallel spins, So, in generala superconducting state in ferromagnetic metal is not equal-spin-pairing state.

Δ0 is relatively small due to smallness of amplitudes V0↑ and Vo↓. And, if we ignore it, this case we deal withtwo-band equal-spin-pairing superconducting state similar to A2

phase of superfluid 3He.

Landau fre energy of orthorhombic ferromagnet

Susceptibilities

Susceptibilities odd components

Pairing amplitudes

Linear GLG equations

Matrix 9x9 differential equation

Superconducting states

The system of equations splits to two independent subsystems corresponding to two differentsuperconducting states with different critical temperature relating to co-representations A and B.

Matrix 5x5 differential equation

Matrix 4x4 differential equation

Let us ignore the small underlined amplitudes corresponding induced pairingwith zero spin projection and study the equal-spin-pairing states.

A

B

A B

Equal-spin-pairing states

A

B

Equal -spin-pairing states near Tc H

T

Hc2(T)

Tc

Hint

A state

B state

Three equations for determination of criticaltemperature in two band superconducting state

Physical properties

Critical temperature

If the largest critical temperature corresponds to superconducting state

Single-band approximation

Upper critical field H||c in UCoGe

T

H

Tc2

Zeros in spectrum

A state B state

In exchange approximation for energy of magnetic inhomogeneity

Zeros on nothern and southern poles Zeros on equator

Zeros on meridional or equatorial lines

Volovik effect

Specific heat at low temperatures

Upper critical field URhGe: Hc2~Tc2

Conventional superconductor

Unconventional superconductor

F.Hardy, A.Huxley, PRL 2005

Upper critical field URhGe: Hc2||c(T)/ Hc2||b(T)=const

F.Hardy, A.Huxley, PRL 2005

Reentrant superconductivity

Reentrant superconductivity phase diagram

Transverse magnetic field plays role similar to pressure.Suppressing the Curie temperature increases pairing amplitude.

First order

12T

F.Levy, I.Sheikin, B.Grenier, A.Huxley (2005)

D.Aoki, G.Knebel, J.Flouquet (2014)

First order phenomenology

I order

II order

PM

FM

Hy

Hy

Mz My

GLG equations

x

yHy

hϕ

There are jumps in (i) band splitting, hence, in density of states,

(ii) in suceptibilities and in angle ϕ, hence, in pairing amplitudes

Field dependence of pairing amplitudes

The longitudinal susceptibilityIn vicinity of the first-ordertransition proves to be muchlarger than the susceptibility atsmall transverse field. This leadsto increase of pairing interactionand stimulates the reentrance ofsuperconductivity.

Jump in Tsc

The Fermi surfaces of split spin-up andspin-down electron bands and the averagedensity of states on them undergo anabrupt change.The structure of equations fordetermination of the critical temperatureof transition in the superconducting stateis quite different on both sides of theferromagnet-paramagnet transition.Hence, the line Tsc(Hy) undergoes a jump atthe intersection of the line fm-pm transition ofthe first order.

URhGe and UCoGe

D.Aoki, J.Flouquet (2014)

a-axis

a - axis is much magnetically harder, hence, anoverturn of magnetization direction is unreachabletill very high magnetic field in a direction.However, an increase of Ha till 30 Tesladoes not kill the reentrant superconductivityexisting near Hb≈12 Tesla.

F.Levy, I.Sheikin and A.huxley, Nat.Phys. 2007

Sharp Hc2angular

dependence

D.Aoki et al 2009

Particular problems

1.Magnitostatics

2. First order phase transition in UGe2

3. Non-Landau damping of magnetic excitationsin systems with localized and itinerant electrons

4. UIr

Magnitostaticsp

f sfs

T

P

x

H

Meissner

T

P

Meissnermixed

1

1H

T

H

3

3

2

2

I Hc2

Hc1

IIHc

T

P

First order phase transition in UGe2

F.Hardy et al 2009 Pfleiderer & Huxley 2002

Fermi gas theory

Abrikosov, Khalatnikov 1958, Kanno 1970Huang, Yang 1957

Fermi liquid theoryBelitz, Kirkpatrick, Vojta 1999

1. In the first order of interaction

2. In the second order of interaction - the transition is of the first order Duine & Macdonald 2005

3. Summation of all the orders of interaction predicts the second order phase transition L.He & X.-G.-Huang, 2012, Monte-Carlo, Pilati 2010

Specific heat jump in UGe2

A.Huxley & SPSMS 2001

A.Huxley et al 2001

m=mU=1.4µB

Striction

Fast drop or growth Tc(P) creates thetendency to the first order transition.

Fluctuations

O.K.Rice 1954

A.I.Larkin &S.A.Pikin 1969

Fluctuation specific heat

A.P.Levanuk 1965

A.I.Larkin & D.E.Khmelnitskii 1969

In UGe2 striction is more important than fluctuations for the phasetransition transformation from the second order type to the first one.

Phase diagram

ZrZn2

10 20 kbar

Uhlarz, Pfleiderer & Hayden 2004

Resume

1. Due to the magneto-elastic interaction the phase transion to the ferromagnet state in UGe2 at

low temperatures is of the first order.

2. A phase transition characterized by the strongly pressuredependent critical temperature is in fact the first orderone.

3. At low temperatures according to the Nernst law and theClausius-Clapeyron relation

the drop of Tc(P) begins to be infinitely fast. It means thata weak first order transition has the tendency to bestronger and stronger as temperature decreases. Hence,the effect of magneto-elastic interaction or, moregenerally, the order parameter interaction with elasticdegrees of freedom at arbitrary type of ordering raisesthe doubts upon existence of quantum criticalphenomena.

Ni, Fe, Co

Sen Yang et al 2008

Non-Landau damping of magneticexcitations in systems with

localized and itinerant electrons

UGe2

R.Troc et al 2012

Neutron scattering

Huxley, Raymond, Ressouche 2003

Van Hove

Line width at T>Tcurie

Diffusion:

G.Shirane et al, 1984

Landau damping:

L.van Hove 1954

B.Halperin, P.Hohenberg 1969

Y.Ishikawa et al, 1982

Mode-mode coupling:

J.Hertz, 1976T.Moriya, 1979

Fe, Ni

MnP, K.Yamada et all 1987, Ni3Al, F.Semadeni et al 2000

MnSi ZrZn2 N.Bernhoeft et al 1988

Line width UGe2, UCoGe

Γq does not vanish as q0 for temperatures different from Tc

Huxley, Raymond, Ressouche, 2003 Stock, Sokolov,….Huxley, 2011

UGe2 localized ferromagnet

← N. Kernavanois et al 2001

A.Yaouanc 2002

Relaxation

Relaxaion above Curie temperature

Landau, Khalatnikov 1954

Relaxation below Curie temperature

At small frequencies

Landau-Lifshitz-Gilbert equations

Line width

RESUME

UIr

Tsc=0.14 KKobayashi et al 2007

UIr superconducting state

a

c

by

References

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