superconductivity in uranium...
TRANSCRIPT
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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